Abstract.

The definition of the complement of a fuzzy subset is algebraic in nature and when it is used in the context of fuzzy topological spaces it does not share any similarity with the usual property of topological spaces that the complement of an open subset is closed. To tackle this inconsistency, we associate to any fuzzy topological space a topological space and use its fundamental groupoid equipped with the Lasso topology to give a topological characterization for the complementation of fuzzy subsets.

1. Introduction and preliminaries

The following is the fuzzy version of the definition of a topological space and appears in [3].

Definition 1.1.

A fuzzy topology is a family $𝒯$ of fuzzy sets in $X$ which satisfies the following conditions:

• $\mathrm{\varnothing },X\in 𝒯$,

• If $A,B\in 𝒯$, then $A\wedge B\in 𝒯$,

• For every family $I$ and every $A_i\in 𝒯$ with $i\in I$, $\underset{i\in I}{\vee }{A}_i\in 𝒯$.

Here $\mathrm{\varnothing }$ is the constant map at 0, and $X$ is the constant map at 1. The mapping $A\wedge B$ is defined by $(A\wedge B)(x)=\text{min}(A(x),B(x))$, and $\underset{i\in I}{\vee }{A}_i$ is defined by $(\underset{i\in I}{\vee }{A}_i)(x)=\underset{i\in I}{\vee }{A}_i(x)$.

This definition generalises that of a topological space since every topological space $(X,\tau )$ can be regarded as a fuzzy topological space in the above sense. Indeed, for every $T\in \tau$, the corresponding indicator map $I_T$ on $X$ is a fuzzy subset of $X$ in the sense of [8], and the collection $𝒯$ of all such $I_T$ is a fuzzy topology on $X$. The indicator map $I_{\theta }$ of the empty map $\theta$ is the constant map $\mathrm{\varnothing }$ at zero, the indicator map $I_X$ is the constant map $X$ at 1, and conditions (ii) and (iii) of definition 1.1 are easily verified. Of course there are examples of fuzzy topological spaces which do not arise from topological spaces in the above manner. One such example has been provided by Lowen in [5] who explores some interesting relationships between topological and fuzzy topological spaces. Lowen associates to any fuzzy topological space $(X,𝒯)$ the topological space $(X,\iota (𝒯))$, where $\iota (𝒯)$ is the initial topology on $X$ for the family of maps in $𝒯$ and the topological space $I_r$. In the reverse direction, he associates to any topological space $(X,\tau )$ the fuzzy topological space $(X,𝒞(X,I_r))$ where $𝒞(X,I_r)$ is the set of all continuous maps from $(X,\tau )$ to $I_r$. It is obvious that the maps of $𝒞(X,I_r)$ are not indicator maps in general, therefore $(X,𝒞(X,I_r))$ serves as a counterexample making thus the difference between topological and fuzzy topological spaces apparent. It should be mentioned that there is another, not so obvious reason, which distinguishes the two notions and has to do with the definition of the complement of a fuzzy subset $F$ of a set $X$ as being the map $1-F$. This definition, which is used widely in the context of fuzzy topological spaces too, is associated with the following pathology. Given any fuzzy topological space $(X,𝒯)$, $T\in 𝒯$ a fuzzy open subset of $X$ and $T^{-1}((\gamma ,1])$ any sub basis member for the topological space $(X,\iota (𝒯))$, the complement of $T^{-1}((\gamma ,1])$ in $(X,\iota (𝒯))$ is in general different than ${(1-T)}^{-1}((\gamma ,1])$ even if $1-T$ is fuzzy open as the following example shows. Let for instance $X$ be any nonempty set and $𝒯=\{1/3,2/3,1,\mathrm{\varnothing }\}$ consisting of four constant maps. If $\gamma =0$ and $T=1/3$, then $1-T=2/3\in 𝒯$ and $T^{-1}((\gamma ,1])=X={(1-T)}^{-1}((\gamma ,1])$, but $X\setminus (T^{-1}((\gamma ,1]))=\theta \ne {(1-T)}^{-1}((\gamma ,1])$. It is now natural to ask the following question. If fuzzy complementary subsets $F$ and $1-F$ in $(X,𝒯)$ do not give rise in general to set theoretic complementary subsets in $(X,\iota (𝒯))$, then is there any other space related to $(X,𝒯)$ in which we can interpret in algebraic or topological terms the process of taking complements of fuzzy subsets in $(X,𝒯)$? The main scope of this paper is to address this question. We leave the details for the definition of that space for the next section, but we have to mention here that our approach is heavily based on the notion of the topological groupoid ${\pi }^{l}Y$ of a given space $Y$ defined from Pakdaman and Shahini in [6]. If $Y$ is a topological space, which is assumed to be connected and locally path connected and $\pi Y$ its fundamental groupoid, Pakdaman and Shahini defined a topology on $\pi Y$ provided that a basis for $Y$ is known. With this topology, the groupoid $\pi Y$ now becomes a topological groupoid and is denoted throughout by ${\pi }^{l}Y$. Pakdaman and Shahini definition is a generalization of the Lasso topology defined in [1] which makes the fundamental group of a space a topological group. It is shown in [6] that a basis for ${\pi }^{l}Y$ ”modulo” a given basis $𝒰$ of $Y$ consists of sets $N([\alpha ],𝒰,V,W)$ with $[\alpha ]\in \pi (Y,x,y)$ where $V,W\in 𝒰$ are open neighbourhoods of $x,y$ respectively, which have elements $[\beta ]$ such that

$$\beta \sim \gamma \ast \mu \ast \alpha \ast {\mu }^{\prime }\ast \lambda ,$$

where $\gamma$ is a path in $V$ with $\gamma (1)=x$, $\mu \in {\pi }_1(𝒰,x)$, ${\mu }^{\prime }\in {\pi }_1(𝒰,y)$ and $\lambda$ is a path in $W$ such that $\lambda (0)=y$. Here ${\pi }_1(𝒰,x)$ is the so called the Spanier group with respect to $𝒰$ and is by definition the subgroup of ${\pi }_1(Y,x)$ consisting of homotopy classes of loops that can be represented by a finite product of the form $\prod _{i\in I}{u}_i{v}_i{u}_i^{-1}$ where $u_i(0)=x$ and $v_i$ is a loop which sits inside some $U_i\in 𝒰$. It is proved in Corollary 2.7 of [6] that there is a functor ${\pi }^l$ from the category ${\mathrm{𝐓𝐨𝐩}}_{\mathrm{𝐜𝐥𝐩𝐜}}$ of connected and locally path connected topological spaces and continuous maps to the category $\mathrm{𝐓𝐨𝐩}\text{-}\mathrm{𝐠𝐫𝐩𝐝}$ of topological groupids. This functor assigns to each space $X$ the corresponding topological groupoid ${\pi }^{l}X$, and maps every morphism $f:X\to Y$ in ${\mathrm{𝐓𝐨𝐩}}_{\mathrm{𝐜𝐥𝐩𝐜}}$ to ${\pi }^l(f):{\pi }^{l}X\to {\pi }^{l}Y$ defined by $[\alpha ]\mapsto [f\circ \alpha ]$. It follows from Proposition 2.6 of [6] that this map is continuous with respect to the lasso topology and therefore it is a morphism in $\mathrm{𝐓𝐨𝐩}\text{-}\mathrm{𝐠𝐫𝐩𝐝}$.

Further we will explain a few notions which are frequently used throughout the paper. If $\alpha :I\to X$ is a path in the space $X$, then its initial point $\alpha (0)$ will be denoted by $\iota (\alpha )$ and its terminal point $\alpha (1)$ by $\tau (\alpha )$. If $\alpha$ is a path in $X$ from $x_0=\alpha (0)$ to $x_1=\alpha (1)$, then its inverse path ${\alpha }^{-1}$ from $x_1$ to $x_0$ is the path defined by ${\alpha }^{-1}(u)=\alpha (1-u)$. If ${\alpha }_1$ and ${\alpha }_2$ are two paths in $X$ such that $\tau ({\alpha }_1)=\iota ({\alpha }_2)$, then ${\alpha }_1\ast {\alpha }_2$ denotes their topological concatenation. By recursion, if ${\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_{n-1},{\alpha }_n$ are paths in $X$ such that for all $1\le i\le n-1$, $\tau ({\alpha }_i)=\iota ({\alpha }_{i+1})$, we use the notation ${\alpha }_1\ast {\alpha }_2\ast \mathrm{\cdots }\ast {\alpha }_{n-1}\ast {\alpha }_n$ to indicate the path $({\alpha }_1\ast {\alpha }_2\ast \mathrm{\cdots }\ast {\alpha }_{n-1})\ast {\alpha }_n$. In the present paper we omit brackets whenever we work via homotopy since the operation $\ast$ is associative up to homotopy. For every space $X$ we denote by $\pi X$ the fundamental groupoid of $X$ whose objects are the elements of $X$ and whose morphisms are the homotopy classes $[\alpha ]$ of paths $\alpha$ in $X$. It is known that $\pi X$ is a groupoid where the composition $[\alpha ]\ast [\beta ]$ of morphisms $[\alpha ]$ and $[\beta ]$ is defined whenever $\alpha \ast \beta$ exists and $[\alpha ]\ast [\beta ]=[\alpha \ast \beta ]$. The inverse ${[\alpha ]}^{-1}$ of $[\alpha ]$ is given by ${[\alpha ]}^{-1}=[{\alpha }^{-1}]$ and the identity morphism $e_x$ at each object $x$ is given by $e_x=[c_x]$ where $c_x$ is the constant path at $x$.

Since the notion of the initial topology on a set induced by a family of maps with domain the given set and codomain a certain topological space is crucial in our paper, we will give its definition as it appears in [4]. Given a set $X$, a topological space $(Y,\tau )$ and a family of maps $f_s:X\to Y$ with $s\in S$, the initial topology $\iota (S)$ on $X$ induced by the family $\{f_s|s\in S\}$ is the coarsest topology on $X$ for which all the maps $f_s$ are continuous. A subbasis for $\iota (S)$ is $𝒮=\{f_s^{-1}(U)\mid s\in S,U\in \tau \}$. Thus the topology $\iota (S)$ consists of all unions of finite intersections of sets from $𝒮$.

Also from [4] we will present here the definition of the product topology on the cartesian product of two spaces together two basic results since some part of the paper uses the product topology and related results. Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces. The product space is the cartesian product $X\times Y$ equipped with the product topology which is the topology generated by the basis $ℬ=\{U\times V\mid U\in {\tau }_X,V\in {\tau }_Y\}$. It follows that an arbitrary open set in $X\times Y$ is a union of such basic open sets. If we let ${\pi }_X:X\times Y\to X$ and ${\pi }_Y:X\times Y\to Y$ be the projection maps in the first and the second coordinate, then it follows that these maps are both open (and continuous). Also for any topological space $Z$ and any mapping $f:Z\to (X\times Y)$, the mapping $f$ is continuous if and only if ${\pi }_X\circ f$ and ${\pi }_Y\circ f$ are both continuous. It follows from this that a map $\alpha :I\to (X\times Y)$ is a path, if and only if ${\pi }_X\circ \alpha$ and ${\pi }_Y\circ \alpha$ are paths in $X$ and $Y$ respectively.

In several auxiliary results in the paper we use the notion of a free homotopy between paths in a space. In order not to get it confused with the notion of a path homotopy, we present it here as it appears in [7]. Let $X$ be a topological space and ${\gamma }_0,{\gamma }_1:I\to X$ be two paths (with possibly different endpoints). A free homotopy between ${\gamma }_0$ and ${\gamma }_1$ is a continuous map $H:I\times I⟶X$ satisfying

$$H(t,0)={\gamma }_0(t),H(t,1)={\gamma }_1(t)\text{ for all }t\in I.$$

We conclude this section by stating that everything which is used in the paper but is not explained in the introduction or latter, can be found in [2] and in [6] if it is related to groupoids, and in [7] if it is related to Algebraic Topology. Necessary notions from fuzzy subsets and fuzzy topological spaces are in [3], [5] and [8].

2. From fuzzy topological spaces to topological spaces

Given a fuzzy topology $𝒯$ on a set $X$, we will define a topology ${\mathrm{\Psi }}^{\ast }(𝒯)$ on the Cartesian product $X\times J$ where $J=[0,1)$ as follows. We first let ${[0,1]}^X$ to be the set of all mapping from $[0,1]$ to $X$ and $𝒫(X\times J)$ the power set of $X\times J$. It is more convenient to define first a map

$${\mathrm{\Psi }}^{\ast }:{[0,1]}^X\to 𝒫(X\times J)$$

such that for every $T\in {[0,1]}^X$,

$${\mathrm{\Psi }}^{\ast }(T)=\{(x,\alpha )\in X\times J|T(x)>\alpha \}.$$

We denote for further use $X_T^{\ast }=\{x\in X|T(x)\ne 0\}$ and note that $P_1({\mathrm{\Psi }}^{\ast }(T))=X_T^{\ast }$, where $P_1$ is the projection in the first coordinate. In the extreme case when $T=\mathrm{\varnothing }$ we see that

$$(x,\alpha )\in {\mathrm{\Psi }}^{\ast }(\mathrm{\varnothing })\iff 0=\mathrm{\varnothing }(x)>\alpha$$

which is impossible, therefore ${\mathrm{\Psi }}^{\ast }(\mathrm{\varnothing })=\mathrm{\varnothing }$. On the other hand, it is clear that for $T\ne \mathrm{\varnothing }$, ${\mathrm{\Psi }}^{\ast }(T)$ can be expressed as a disjoint union of sets as follows

$${\mathrm{\Psi }}^{\ast }(T)=\bigcup _{x\in X_T^{\ast }}\{x\}\times [0,T(x)),$$ (1)

and that for every fixed $x\in X_T^{\ast }$,

$$T(x)=\text{sup}\{\alpha \in J|(x,\alpha )\in {\mathrm{\Psi }}^{\ast }(T)\}.$$ (2)

Let us show that if we restrict ${\mathrm{\Psi }}^{\ast }$ on $𝒯$, then the respective family of subsets ${\mathrm{\Psi }}^{\ast }(𝒯)=\{{\mathrm{\Psi }}^{\ast }(T)|T\in 𝒯\}$ constitutes to a topology on $X\times J$. Indeed, as we previously observed, ${\mathrm{\Psi }}^{\ast }(\mathrm{\varnothing })=\mathrm{\varnothing }$, and so $\mathrm{\varnothing }\in {\mathrm{\Psi }}^{\ast }(𝒯)$. If we take $T=X$, then

$(x,\alpha )\in {\mathrm{\Psi }}^{\ast }(X)$ $\iff 1=X(x)>\alpha$

$\iff (x,\alpha )\in X\times J,$

which shows that $X\times J\in {\mathrm{\Psi }}^{\ast }(𝒯)$. Now let $T_1,T_2\in 𝒯$. We show that

$${\mathrm{\Psi }}^{\ast }(T_1)\cap {\mathrm{\Psi }}^{\ast }(T_2)={\mathrm{\Psi }}^{\ast }(T_1\wedge T_2)$$ (3)

from which we get that ${\mathrm{\Psi }}^{\ast }(T_1\wedge T_2)\in {\mathrm{\Psi }}^{\ast }(𝒯)$ since $T_1\wedge T_2\in 𝒯$. This proves that ${\mathrm{\Psi }}^{\ast }(𝒯)$ is closed under finite intersections. Indeed, if $T_1\wedge T_2=\mathrm{\varnothing }$, then there are no pairs $(x,\alpha )\in {\mathrm{\Psi }}^{\ast }(T_1)\cap {\mathrm{\Psi }}^{\ast }(T_2)$, therefore

$${\mathrm{\Psi }}^{\ast }(T_1)\cap {\mathrm{\Psi }}^{\ast }(T_2)=\mathrm{\varnothing }={\mathrm{\Psi }}^{\ast }(T_1\wedge T_2).$$

If $T_1\wedge T_2\ne \mathrm{\varnothing }$, then for some $x\in X$, $T_1(x)\wedge T_2(x)\ne 0$, consequently ${\mathrm{\Psi }}^{\ast }(T_1)\cap {\mathrm{\Psi }}^{\ast }(T_2)\ne \mathrm{\varnothing }$ since for every we have that $(x,\alpha )\in {\mathrm{\Psi }}^{\ast }(T_1)\cap {\mathrm{\Psi }}^{\ast }(T_2)$. Further, to complete the proof for (3) in the latter case, we see that

$(x,\alpha )\in {\mathrm{\Psi }}^{\ast }(T_1)\cap {\mathrm{\Psi }}^{\ast }(T_2)$ $\iff T_1(x)>\alpha \text{ and }{T}_2(x)>\alpha$

$\iff (T_1(x)\wedge T_2(x))>\alpha$

$\iff (T_1\wedge T_2)(x)>\alpha$

$\iff (x,\alpha )\in {\mathrm{\Psi }}^{\ast }(T_1\wedge T_2).$

Finally, let $I$ be any nonempty index set and let $T_i\in 𝒯$ for $i\in I$. We show that

$$\underset{i\in I}{\cup }{\mathrm{\Psi }}^{\ast }(T_i)={\mathrm{\Psi }}^{\ast }(\underset{i\in I}{\vee }(T_i)),$$ (4)

from which we get that ${\mathrm{\Psi }}^{\ast }(\underset{i\in I}{\vee }(T_i))\in {\mathrm{\Psi }}^{\ast }(𝒯)$ since $\underset{i\in I}{\vee }{T}_i\in 𝒯$. This proves that ${\mathrm{\Psi }}^{\ast }(𝒯)$ is closed under arbitrary unions. Indeed, if each $T_i=\mathrm{\varnothing }$, then the equality is clear. Otherwise, $\underset{i\in I^{\prime }}{\cup }{\mathrm{\Psi }}^{\ast }(T_i)$ is nonempty in which case we see that

$(x,\alpha )\in \underset{i\in I}{\cup }{\mathrm{\Psi }}^{\ast }(T_i)$ $\iff \exists i\in I,(x,\alpha )\in {\mathrm{\Psi }}^{\ast }(T_i)$

$\iff \exists i\in I,T_i(x)>\alpha$

$\iff \underset{i\in I}{\vee }{T}_i(x)>\alpha$

$\iff (x,\alpha )\in {\mathrm{\Psi }}^{\ast }(\underset{i\in I}{\vee }{T}_i).$

The problem with the topology ${\mathrm{\Psi }}^{\ast }(𝒯)$ on $X\times J$ defined in terms of the fuzzy topology $𝒯$ on $X$, is that the correspondence ${\mathrm{\Psi }}^{\ast }$ does not extends to closed subsets. More specifically, it may happen that for some fuzzy open subset $T\in 𝒯$, we have that ${\mathrm{\Psi }}^{\ast }(1-T)\ne (X\times J)\setminus {\mathrm{\Psi }}^{\ast }(T)$ as the subsequent example shows. In contrast with this, when $T$ is the indicator map $I_A$ of some $A\subseteq X$, we have that ${\mathrm{\Psi }}^{\ast }(1-T)=(X\times J)\setminus {\mathrm{\Psi }}^{\ast }(T)$. Indeed, when $A=X$, then $I_A=1$ the constant map at 1 and $1-I_A=\theta$ the constant map at zero. On the other hand ${\mathrm{\Psi }}^{\ast }(1)=X\times J$ and ${\mathrm{\Psi }}^{\ast }(\theta )=\mathrm{\varnothing }$. In this case we see that ${\mathrm{\Psi }}^{\ast }(1-I_A)=(X\times J)\setminus {\mathrm{\Psi }}^{\ast }(I_A)$. The case when $A=\mathrm{\varnothing }$ is dual to the above. When $A\subset X$ is a nontrivial subset, we observe that $1-I_A=I_{X\setminus A}$ and that

${\mathrm{\Psi }}^{\ast }(1-I_A)$ $=\{(x,\alpha )\in X\times J:I_{X\setminus A}(x)>\alpha \}$

$=(X\setminus A)\times J$

$=(X\times J)\setminus (A\times J)$

$=(X\times J)\setminus \{(x,\alpha )\in X\times J:I_A(x)>\alpha \}$

$=(X\times J)\setminus {\mathrm{\Psi }}^{\ast }(I_A).$

Example 2.1.

Let $X$ be any nonempty set and let $𝒯=\{\theta ,1,{\displaystyle \frac{1}{3}}\}$ be the fuzzy topology on $X$ consisting of only these three constant maps. Letting $T={\displaystyle \frac{1}{3}}$ we see that

$${\mathrm{\Psi }}^{\ast }(T)=\{(x,\alpha )\in X\times J:\frac{1}{3}>\alpha \}=X\times [0,1/3[,$$

and then

$$(X\times J)\setminus {\mathrm{\Psi }}^{\ast }(T)=X\times [1/3,1[.$$

But

$${\mathrm{\Psi }}^{\ast }(1-T)=\{(x,\alpha )\in X\times J:1-1/3>\alpha \}=X\times [0,2/3[,$$

which shows that ${\mathrm{\Psi }}^{\ast }(1-T)\ne (X\times J)\setminus {\mathrm{\Psi }}^{\ast }(T)$.

This oddity that steams from the definition of fuzzy complements which is algebraic in nature, motivates us to construct another, more complex topological space in which taking complements of fuzzy subsets on the one side, is equivalent on the other side with taking inverses. This idea has led us to consider the groupoid of homotopy classes of paths on $X\times J$ but the topology ${\mathrm{\Psi }}^{\ast }(𝒯)$ is to coarse to achieve anything, therefore it is tempting to refine ${\mathrm{\Psi }}^{\ast }(𝒯)$ in a reasonable way. Our approach in doing this is based on Lowen’s idea in [5] who associates any given fuzzy topology $𝒯$ on $X$ with the initial topology on $X$ for the family of maps $T\in 𝒯$ and the topological space $I_r$.

Definition 2.2.

Consider the set $(-1,1]$ and the family of all left intervals $(\gamma ,1]$ with . We let ${(-1,1]}_r$ be the topological space generated by this family.

Definition 2.3.

For every $T\in 𝒯$ we define $T^{\ast }:X\times J\to {(-1,1]}_r$ by $T^{\ast }(x,\alpha )=T(x)-\alpha$.

Notation 2.4.

Let $P_2:X\times J\to {(-1,1]}_r$ be the projection in the second coordinate, $P_2((x,\alpha ))=\alpha$.

Notation 2.5.

Let $\iota (𝒯)$ be the initial topology on $X\times J$ for the family of maps $T^{\ast }$ with $T\in 𝒯$ together with $P_2$, and the topological space ${(-1,1]}_r$.

Lemma 2.6.

The topology $\iota (𝒯)$ is finer than ${\mathrm{\Psi }}^{\ast }(𝒯)$.

Proof 2.7.

For every $T\in 𝒯$, we see that ${(T^{\ast })}^{-1}((0,1])=\{(x,\alpha )\in X\times J:T(x)-\alpha \in (0,1]\}=\{(x,\alpha )\in X\times J:T(x)>\alpha \}={\mathrm{\Psi }}^{\ast }(T)$.

Notation 2.8.

Let ${\iota }_X(𝒯)$ be the initial topology on $X$ for the family of maps $T\in 𝒯$ and the topological space ${(-1,1]}_r$.

Lemma 2.9.

Every $T\in 𝒯$ is a continuous map from the space $(X,{\iota }_X(𝒯))$ to ${(-1,1]}_r$.

Proof 2.10.

This follows from the general result that the initial topology on a family of maps makes these maps continuous.

Remark 2.11.

A sub basis for $\iota (𝒯)$ consists in all ${(T^{\ast })}^{-1}((\gamma ,1])$ with $T\in 𝒯$ and together with ${(P_2)}^{-1}((\gamma ,1])$ with . To make the second more explicit, we see that ${(P_2)}^{-1}((\gamma ,1])=\{(x,\alpha )\in X\times J:\alpha >\gamma \}$. If we consider now the subspace $X\times \{0\}$ of $(X\times J,\iota (𝒯))$ where the topology is the one inherited from $\iota (𝒯)$, we see that for every , ${(P_2)}^{-1}((\gamma ,1])=\{(x,0)\in X\times \{0\}:0>\gamma \}=X\times \{0\}$, and for every $\gamma \ge 0$, ${(P_2)}^{-1}((\gamma ,1])=\{(x,0)\in X\times \{0\}:0>\gamma \}=\mathrm{\varnothing }$. Therefore the only sub basis member for $(X\times \{0\},\iota (𝒯))$ arising from $P_2$ is $X\times \{0\}$ itself. Finally, we mention that it follows from the definition ${\iota }_X(𝒯)$ that a sub basis for ${\iota }_X(𝒯)$ consist of all subsets $T^{-1}((\gamma ,1])$ with $T\in 𝒯$ and .

Lemma 2.12.

The projection in the first coordinate $P_1:X\times \{0\}\to X$ is a homeomorphism of spaces.

Proof 2.13.

Let $(x,0)\in X\times \{0\}$ and $T^{-1}((\gamma ,1])$ an open sub basis member containing $x=P_1(x,0)$. It follows that $T(x)>\gamma$, whence $(x,0)\in {(T^{\ast })}^{-1}(\gamma ,1]$. For every $(y,0)\in {(T^{\ast })}^{-1}((\gamma ,1])$, since $T(y)>\gamma$, we have that $y\in T^{-1}((\gamma ,1])$ which proves that $P_1$ is continuous. Conversely, let $(x,0)=P_1^{-1}(x)$ and let ${(T^{\ast })}^{-1}((\gamma ,1])$ be an open sub basis neighbourhood of $(x,0)$. Since $T(x)>\gamma$, we have that $x\in T^{-1}((\gamma ,1])$. For every $y\in T^{-1}((\gamma ,1])$, $T(y)>\gamma$, hence $P_1^{-1}(y)=(y,0)\in {(T^{\ast })}^{-1}((\gamma ,1])$ proving that $P_1^{-1}$ is also continuous.

Theorem 2.14.

The space $(X\times J,\iota (𝒯))$ deformation retracts to $(X\times \{0\},\iota (𝒯))$.

Proof 2.15.

Let $H:I\times (X\times J)\to X\times J$ defined by $H(t,(x,\alpha ))=(x,(1-t)\alpha )$. First we note that $H(t,(x,0))=(x,0)$ for every $(x,0)\in X\times \{0\}$, $H(0,(x,\alpha ))=(x,\alpha )$ for all $(x,\alpha )\in X\times J$ and $H(1,(x,\alpha ))=(x,0)$ for all $(x,\alpha )\in X\times J$. Now we prove that $H$ is continuous by testing it on the sub basis described in the previous remark. We distinguish between the following cases for $t$.
(i) $t=1$. Let $P_2^{-1}((\gamma ,1])$ with be an open neighbourhood of $(x,0)=(x,(1-1)\alpha )$. It follows that . If $\alpha =0$, then for some appropriately chosen $\epsilon >0$, $(1-\epsilon ,1]\times P_2^{-1}((\gamma ,1])$ is an open neighbourhood of $(1,(x,0))$. For every $(1-\delta ,(y,\beta ))\in (1-\epsilon ,1]\times P_2^{-1}((\gamma ,1])$ we have that $H((1-\delta ,(y,\beta )))=(y,\delta \beta )\in P_2^{-1}((\gamma ,1])$ since $\delta \beta \ge 0>\gamma$. In the case when $\alpha >0$, we can chose $\epsilon >0$ such that $1-\epsilon >0$ and $\alpha -\epsilon >0$. It follows that an open neighbourhood of $(1,(x,\alpha ))$ is $(1-\epsilon ,1]\times P_2^{-1}((\alpha -\epsilon ,1])$. For every $(1-\delta ,(y,\beta ))\in (1-\epsilon ,1]\times P_2^{-1}((\alpha -\epsilon ,1])$ we have that $H((1-\delta ,(y,\beta )))=(y,\delta \beta )\in P_2^{-1}((\gamma ,1])$ since $\delta \beta \ge \delta (\alpha -\epsilon )\ge 0>\gamma$. In the case when ${(T^{\ast })}^{-1}((\gamma ,1])$ with is an open neighbourhood of $(x,0)=(x,(1-1)\alpha )$, we have that $T(x)>\gamma$. If $\alpha =0$, then an open neighbourhood of $(1,(x,0))$ is $(1-\epsilon ,1]\times {(T^{\ast })}^{-1}((\gamma ,1])$ where . For every $(1-\delta ,(y,\beta ))\in (1-\epsilon ,1]\times {(T^{\ast })}^{-1}((\gamma ,1])$ we have that $H((1-\delta ,(y,\beta )))=(y,\delta \beta )\in {(T^{\ast })}^{-1}((\gamma ,1])$. Indeed,

$T(y)$ $>\gamma +\beta$ $(\text{from the assumption that }(y,\beta )\in {(T^{\ast })}^{-1}((\gamma ,1]))$

$\ge \gamma +\delta \beta$

which means that $T(y)-\delta \beta \in (\gamma ,1]$, or equivalently that $H((1-\delta ,(y,\beta )))=(y,\delta \beta )\in {(T^{\ast })}^{-1}((\gamma ,1])$. In case $\alpha >0$, we can chose such that $T(x)-\alpha >\gamma -(\alpha -2\epsilon )$ and $\alpha -\epsilon >0$. An open neighbourhood of $(1,(x,\alpha ))$ now is $(1-\epsilon ,1]\times ({(T^{\ast })}^{-1}((\gamma -(\alpha -2\epsilon )]\cap P_2^{-1}(\alpha -\epsilon ,1])$. Let $(1-\delta ,(y,\beta ))\in (1-\epsilon ,1]\times ({(T^{\ast })}^{-1}((\gamma -(\alpha -2\epsilon )]\cap P_2^{-1}(\alpha -\epsilon ,1])$, then $T(y)-\beta >\gamma -(\alpha -2\epsilon )$. It follows that for every $1-\delta \in (1-\epsilon ,1]$ we have

$T(y)-(1-(1-\delta ))\beta$ $=T(y)-\beta +(1-\delta )\beta$

$>\gamma -(\alpha -2\epsilon )+(1-\delta )(\alpha -\epsilon )$

$=\gamma -(\alpha -\epsilon )+\epsilon +(\alpha -\epsilon )-\delta (\alpha -\epsilon )$

$=\gamma +\epsilon -\delta (\alpha -\epsilon )$

$\ge \gamma$ $(\text{since }\epsilon -\delta (\alpha -\epsilon )\ge 0)$

(ii) . As in the previous case, an open neighbourhood of $(x,(1-t)\alpha )$ can be of two kinds. The first one is $P_2^{-1}((\gamma ,1])$ with . Then, $\alpha >\gamma +t\alpha$. We can find $\epsilon >0$ such that $(t-\epsilon ,t+\epsilon )\subset [0,1]$ and $\alpha >\gamma +(t+\epsilon )\alpha$. It follows that $(x,\alpha )\in P_2^{-1}((\gamma +(t+\epsilon )\alpha ,1])$, and that an open neighbourhood of $(t,(x,\alpha ))$ is $(t-\epsilon ,t+\epsilon )\times P_2^{-1}((\gamma +(t+\epsilon )\alpha ,1])$. We prove that $H\left((t-\epsilon ,t+\epsilon )\times P_2^{-1}((\gamma +(t+\epsilon )\alpha ,1])\right)\subseteq P_2^{-1}((\gamma ,1])$. Let $(t+\delta ,(y,\beta ))\in (t-\epsilon ,t+\epsilon )\times P_2^{-1}((\gamma +(t+\epsilon )\alpha ,1])$, hence and $\beta >\gamma +(t+\epsilon )\alpha$. For $H((t+\delta ,(y,\beta )))=(y,(1-(t+\delta ))\beta )$ we see that

$(1-(t+\delta ))\beta$ $\ge (1-(t+\epsilon ))\beta$

$>(1-(t+\epsilon ))(\gamma +(t+\epsilon )\alpha )$

$=\gamma +(t+\epsilon )(\alpha -\gamma -(t+\epsilon )\alpha )$

$\ge \gamma$ $(\text{since }\alpha >\gamma +(t+\epsilon )\alpha ).$

This proves that $H((t+\delta ,(y,\beta )))\in P_2^{-1}((\gamma ,1])$. The second kind of an open neighbourhood of $(x,(1-t)\alpha )$ is ${(T^{\ast })}^{-1}((\gamma ,1])$ with . In this case $T(x)-\alpha >\gamma -\alpha t$. We can chose $\epsilon >0$ such that $(t-\epsilon ,t+\epsilon )\subset [0,1]$ and that

$$T(x)-\alpha >\gamma -\alpha t+(\epsilon t+\epsilon )=\gamma -(\alpha -(\epsilon +\frac{\epsilon }{t}))t.$$

Let $\mu =\text{max}\{-1,\gamma -(\alpha -(\epsilon +{\displaystyle \frac{\epsilon }{t}}))t\}$. It follows that $(x,\alpha )\in {(T^{\ast })}^{-1}((\mu ,1])\cap P_2^{-1}((\alpha -\epsilon ,1])$. Thus an open neighborhood of $(t,(x,\alpha ))$ is $(t-\epsilon ,t+\epsilon )\times ({(T^{\ast })}^{-1}((\mu ,1])\cap P_2^{-1}((\alpha -\epsilon ,1]))$. We prove that $H((t-\epsilon ,t+\epsilon )\times ({(T^{\ast })}^{-1}((\mu ,1])\cap P_2^{-1}((\alpha -\epsilon ,1])))\subseteq {(T^{\ast })}^{-1}((\gamma ,1])$. Indeed, for every $(t+\delta ,(y,\beta ))\in (t-\epsilon ,t+\epsilon )\times ({(T^{\ast })}^{-1}((\mu ,1])\cap P_2^{-1}((\alpha -\epsilon ,1]))$, we have that , $T(y)-\beta >\mu \ge \gamma -(\alpha -(\epsilon +{\displaystyle \frac{\epsilon }{t}}))t$ and $\beta >\alpha -\epsilon$. It follows that

$T(y)-(1-(t+\delta ))\beta$ $=T(y)-\beta +t\beta +\delta \beta$

$>\gamma -(\alpha -(\epsilon +{\displaystyle \frac{\epsilon }{t}}))t+t(\alpha -\epsilon )+\delta \beta$

$=\gamma -(\alpha -\epsilon )t+\epsilon +t(\alpha -\epsilon )+\delta \beta$

$=\gamma +\epsilon +\delta \beta$

$\ge \gamma$ $(\text{since }\epsilon >\beta \epsilon >\beta |\delta |,)$

which show that $H((t+\delta ,(y,\beta )))\in {(T^{\ast })}^{-1}((\gamma ,1])$.
(iii) $t=0$. Let $P_2^{-1}((\gamma ,1])$ be an open neighbourhood of $H(0,(x,\alpha ))=(x,\alpha )$, hence $\alpha >\gamma$. When $\alpha \ne 0$, we can find such that $\alpha >{\displaystyle \frac{1}{1-\epsilon }}\gamma$. We can in fact chose . It is clear that $(0,(x,\alpha ))\in [0,\epsilon )\times P_2^{-1}(({\displaystyle \frac{1}{1-\epsilon }}\gamma ,1])$. Let now $(\delta ,(y,\beta ))\in [0,\epsilon )\times P_2^{-1}(({\displaystyle \frac{1}{1-\epsilon }}\gamma ,1])$, hence and $\beta >{\displaystyle \frac{1}{1-\epsilon }}\gamma$. It follows that $H((\delta ,(y,\beta )))=(y,(1-\delta )\beta )\in P_2^{-1}((\gamma ,1])$ since

$$\beta >\frac{1}{1-\epsilon }\gamma \ge \frac{1}{1-\delta }\gamma ,$$

which implies that $(1-\delta )\beta >\gamma$. Now assume that $\alpha =0$, then $H(0,(x,0))=(x,0)\in P_2^{-1}((\gamma ,1])$. This means that . For every , $(0,(x,0))\in [0,\epsilon )\times P_2^{-1}((\gamma ,1])$. Let $(\delta ,(y,\beta ))\in [0,\epsilon )\times P_2^{-1}((\gamma ,1])$, then and $\beta >\gamma$. It follows that $(1-\delta )\beta >(1-\delta )\gamma \ge \gamma$, therefore $H((\delta ,(y,\beta )))=(y,(1-\delta )\beta )\in P_2^{-1}((\gamma ,1])$. Finally assume that $(0,(x,\alpha ))\in {(T^{\ast })}^{-1}((\gamma ,1])$. Since $T(x)-\alpha >\gamma$, then for every , $[0,\epsilon )\times {(T^{\ast })}^{-1}((\gamma ,1])$ is an open neighbourhood of $(0,(x,\alpha ))$. For every $(\delta ,(y,\beta ))\in [0,\epsilon )\times {(T^{\ast })}^{-1}((\gamma ,1])$ we see that

$$T(y)-(1-\delta )\beta \ge T(y)-\beta >\gamma .$$

This proves that $H((\delta ,(y,\beta )))=(y,(1-\delta )\beta )\in {(T^{\ast })}^{-1}((\gamma ,1])$, concluding the proof.

3. The groupoid approach

In this section we will consider the topological groupoid ${\pi }^l(X\times J)$ of homotopy classes of paths on $X\times J$ equipped with the Lasso topology. The topological groupoid ${\pi }^l(Y)$ whose elements are homotopy classes of paths on $Y$ has been defined in [6] under the assumption that $Y$ is path connected (p.c) and locally path connected (l.p.c), therefore the definition of our ${\pi }^l(X\times J)$ would require $X\times J$ to be p.c and l.p.c. It turns out from Proposition 3.11 that in order for $X\times J$ to be p.c and l.p.c., it is enough to assume that $(X,{\iota }_X(𝒯))$ is p.c and l.p.c. The first part of this section is devoted to the proof of Proposition 3.11. In order to make the proof easy to follow we have proved in advance a number of technical results one of which is Lemma 3.7 that uses Remark 3.23, a result which belongs to the second part of the section. The second part contains also a few other technical results that precede the proof of Theorem 3.30 which states that ${\pi }^l(X\times J)$ deformation retract onto ${\pi }^l(X\times \{0\})$. The result of Theorem 3.30 will be essential in section 4.

In what follows we will let $\sigma :X\times J\to X\times \{0\}$ be the restriction of $H$ on $\{1\}\times (X\times J)$. Since $H$ is a deformation retraction, $\sigma$ has to be a homotopy equivalence. Here the topology in $X\times \{0\}$ is the one derived from that in $X\times J$.

Lemma 3.1.

For every path $p$ in $X\times \{0\}$ and every fixed $\alpha \in J$, we let ${}_{\alpha }{}^{}p:I\to X\times J$ defined by ${}_{\alpha }{}^{}p(u)=(P_1(p(u)),\alpha )$ where $P_1:X\times \{0\}\to X$ is the homeomorphism of Lemma 2.12. The map ${}_{\alpha }{}^{}p$ is a path in $X\times J$.

Proof 3.2.

If ${(T^{\ast })}^{-1}((\gamma ,1])$ is a sub basis member containing $(P_1(p(u)),\alpha )$, then $(T{P}_{1}p)(u)>\alpha +\gamma$. But the map $T:X\to I$ is continuous, therefore ${(T{P}_{1}p)}^{-1}((\alpha +\gamma ,1]\cap I)$ is an open subset of $I$ and it contains $u$. For every $v\in {(T{P}_{1}p)}^{-1}((\alpha +\gamma ,1]\cap I)$, $(T{P}_{1}p)(v)>\alpha +\gamma$, which means that ${}_{\alpha }{}^{}p(v)=(P_1(p(v)),\alpha )\in {(T^{\ast })}^{-1}((\gamma ,1])$. If now a sub basis member containing $(P_1(p(u)),\alpha )$ is $P_2^{-1}((\gamma ,1])$, then $\alpha >\gamma$. In this case as an open neighbourhood of $\alpha$ we can take the whole of $I$ since for every $v\in I$, ${}_{\alpha }{}^{}p(v)=(P_1(p(v)),\alpha )\in P_2^{-1}((\gamma ,1])$. It follows that ${}_{\alpha }{}^{}p$ is continuous.

Lemma 3.3.

For every finite number of open subsets of the form ${(T_i^{\ast })}^{-1}(({\gamma }_i,1])$, we have that
$\sigma (\underset{𝑖}{\cap }{(T_i^{\ast })}^{-1}(({\gamma }_i,1]))=\underset{𝑖}{\cap }\sigma ({(T_i^{\ast })}^{-1}(({\gamma }_i,1]))$.

Proof 3.4.

We prove the non obvious inclusion $\underset{𝑖}{\cap }\sigma ({(T_i^{\ast })}^{-1}(({\gamma }_i,1]))\subseteq \sigma (\underset{𝑖}{\cap }{(T_i^{\ast })}^{-1}(({\gamma }_i,1]))$. Let $(x,0)\in \underset{𝑖}{\cap }\sigma ({(T_i^{\ast })}^{-1}(({\gamma }_i,1]))$, then there are ${\alpha }_i\in J$ such that $T_i(x)>{\alpha }_i+{\gamma }_i\ge {\gamma }_i$. This implies that $(x,0)\in \underset{𝑖}{\cap }{(T_i^{\ast })}^{-1}(({\gamma }_i,1]))$. Since $(x,0)=\sigma (x,0)$, the inclusion follows.

Lemma 3.5.

For every open subset of the form ${(T^{\ast })}^{-1}((\gamma ,1])$ in $X\times J$, every $(y,\beta )\in {(T^{\ast })}^{-1}((\gamma ,1])$ and every $0\le {\beta }^{\prime }\le \beta$, $(y,{\beta }^{\prime })\in {(T^{\ast })}^{-1}((\gamma ,1])$. In particular, if $(y,\beta )\in {(T^{\ast })}^{-1}((\gamma ,1])$, then $(y,0)\in {(T^{\ast })}^{-1}((\gamma ,1])$.

Proof 3.6.

Indeed, since $T(y)>\beta +\gamma$, then $T(y)>{\beta }^{\prime }+\gamma$, hence $(y,{\beta }^{\prime })\in {(T^{\ast })}^{-1}((\gamma ,1])$.

Lemma 3.7.

• If $(y,{\beta }_1),(y,{\beta }_2)\in {(T^{\ast })}^{-1}((\gamma ,1])$, then the path $p$ arising from Remark 3.23 which connects $(y,{\beta }_1)$ with $(y,{\beta }_2)$ sits inside ${(T^{\ast })}^{-1}((\gamma ,1])$.

• If $(y,{\beta }_1),(y,{\beta }_2)\in P_2^{-1}((\gamma ,1])$, then the path $p$ arising from Remark 3.23 which connects $(y,{\beta }_1)$ with $(y,{\beta }_2)$ sits inside $P_2^{-1}((\gamma ,1])$.

Proof 3.8.

(i) This follows from Lemma 3.5 since each point of $p$ is a pair $(y,\beta )$ with ${\beta }_1\le \beta \le {\beta }_2$.
(ii) Indeed, any point of $p$ is a pair $(y,\beta )$ with ${\beta }_1\le \beta \le {\beta }_2$. But ${\beta }_1>\gamma$, hence $\beta >\gamma$ and $(y,\beta )\in P_2^{-1}((\gamma ,1])$.

Lemma 3.9.

The images under $\sigma$ of sub basis members of $\iota (𝒯)$ are open in $X\times \{0\}$.

Proof 3.10.

First we assume that $B={(T^{\ast })}^{-1}((\gamma ,1])$, with $T\in 𝒯$ and . For every $(x,0)=\sigma (x,\alpha )$ with $(x,\alpha )\in {(T^{\ast })}^{-1}((\gamma ,1])$ we have that $T(x)>\alpha +\gamma \ge \gamma$ from which we get that $(x,0)\in {(T^{\ast })}^{-1}((\gamma ,1])\cap (X\times \{0\})$. So we have in this case that

$$\sigma ({(T^{\ast })}^{-1}((\gamma ,1]))\subseteq {(T^{\ast })}^{-1}((\gamma ,1])\cap (X\times \{0\}).$$

But the converse inclusion holds true as well for if $(x,0)\in {(T^{\ast })}^{-1}((\gamma ,1])\cap (X\times \{0\})$, then $(x,0)=\sigma (x,0)$. Therefore we have that

$$\sigma ({(T^{\ast })}^{-1}((\gamma ,1]))={(T^{\ast })}^{-1}((\gamma ,1])\cap (X\times \{0\}),$$

and ${(T^{\ast })}^{-1}((\gamma ,1])\cap (X\times \{0\})$ is open in the subspace topology. Secondly, if the sub basis member is $(P_2^{-1})((\gamma ,1])$, where we can assume without loss of generality that $\gamma \ge 0$, then it follows that $\sigma (P_2^{-1}((\gamma ,1]))=X\times \{0\}$ since for $\gamma \ge 0$, $P_2^{-1}((\gamma ,1])=X\times (\gamma ,1]$, hence $\sigma (P_2^{-1}((\gamma ,1]))$ is open.

Proposition 3.11.

• $X\times \{0\}$ is l.p.c., if and only if $X\times J$ is l.p.c.

• $X\times \{0\}$ is p.c., if and only if $X\times J$ is p.c.

Proof 3.12.

(i) Assume that $X\times \{0\}$ is l.p.c. Let $(x,\alpha )$ be arbitrary and $U$ some open subset of $X\times J$ containing $(x,\alpha )$. Then $U$ decomposes as $U=\underset{i\in ℐ}{\cup }{C}_i$ where each $C_i=\underset{j\in {𝒥}_i}{\cap }{B}_{i,j}$ with ${𝒥}_i$ finite and each $B_{i,j}$ being a sub basis member. It follows that for some $i\in ℐ$, $(x,\alpha )\in C_i$. There are two possible cases for $C_i$. In the first one, all the intersecting components $B_{i,j}$ of $C_i$ are of the form $P_2^{-1}(({\gamma }_{i,j},1])$, hence $C_i=P_2^{-1}(({\gamma }_i,1])$ where ${\gamma }_i=\text{max}\{{\gamma }_{i,j}:j\in {𝒥}_i\}$. Lemma 3.7, (ii) implies that there is a path connecting any two points $(x_1,{\alpha }_1),(x_2,{\alpha }_2)\in P_2^{-1}(({\gamma }_i,1])$ which is included in $P_2^{-1}(({\gamma }_i,1])$ and therefore in $U$. The second case for $C_i$ is when all of the $B_{i,j}$ are of the form ${(T_1^{\ast })}^{-1}(({\gamma }_1,1]),\mathrm{\dots },{(T_n^{\ast })}^{-1}(({\gamma }_n,1])$. From Lemma 3.3 and Lemma 3.9, $\sigma ({(T_1^{\ast })}^{-1}(({\gamma }_1,1]))\cap \mathrm{\dots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n,1]))$ is an open neighbourhood of $(x,0)$. From the assumption on $X\times \{0\}$, there is neighbourhood $V^{\ast }$ of $(x,0)$ such that $V^{\ast }\subseteq \sigma ({(T_1^{\ast })}^{-1}(({\gamma }_1,1]))\cap \mathrm{\dots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n,1]))$ with the property that any two points in $V^{\ast }$ can be connected by a path in $\sigma ({(T_1^{\ast })}^{-1}({\gamma }_1,1])\cap \mathrm{\dots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n,1]))$. Since $\sigma$ is continuous, ${\sigma }^{-1}(V^{\ast })$ is an open neighbourhood of $(x,\alpha )$, hence

$${\sigma }^{-1}(V^{\ast })\cap {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])$$

is also an open neighbourhood of $(x,\alpha )$ and

$${\sigma }^{-1}(V^{\ast })\cap {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\subseteq {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1]).$$

Let now $(x_1,{\alpha }_1),(x_2,{\alpha }_2)\in {\sigma }^{-1}(V^{\ast })\cap {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])$. It follows for their images under $\sigma$ that

$(x_1,0),(x_2,0)$ $\in \sigma ({\sigma }^{-1}(V^{\ast })\cap {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1]))$

$\subseteq V^{\ast }\cap \sigma ({(T_1^{\ast })}^{-1}(({\gamma }_1,1]))\cap \mathrm{\dots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n,1]))$

$=V^{\ast }.$

From our assumption, there is a path $p^{\ast }$ connecting $(x_1,0)$ with $(x_2,0)$ which is included in $\sigma ({(T_1^{\ast })}^{-1}({\gamma }_1,1])\cap \mathrm{\dots }\cap \sigma ({(T_n^{\ast })}^{-1}({\gamma }_n,1])$. But

$$\sigma ({(T_1^{\ast })}^{-1}(({\gamma }_1,1]))\cap \mathrm{\dots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n,1]))\subseteq {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1]),$$

hence $p^{\ast }$ is also included in ${(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])$. From Lemma 3.7,(i) there are paths, $q_1$ connecting $(x_1,{\alpha }_1)$ with $(x_1,0)$, and $q_2$ connecting $(x_2,{\alpha }_2)$ with $(x_2,0)$, which are inside ${(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])$. It follows that $q_1\ast p^{\ast }\ast q_2^{-1}$ is a path that connects $(x_1,{\alpha }_1)$ to $(x_2,{\alpha }_2)$ which sits inside ${(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\dots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])$.

The third case for $C_i$ is when some of the $B_{i,j}$ are of the form ${(T_1^{\ast })}^{-1}(({\gamma }_1,1]),\mathrm{\dots },{(T_n^{\ast })}^{-1}(({\gamma }_n,1])$, and the rest of the components form a non-empty family consisting of open sets $P_2^{-1}({\delta }_j,1]$ whose intersection has the form $P_2^{-1}((\delta ,1])$, therefore

$$C_i={(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1]).$$

Since $(x,\alpha )\in C_i$ we have that $\alpha >\delta$ and for every $i=1,\mathrm{\dots },n$, $T_i(x)>{\gamma }_i+\alpha$ which means in particular that

$$(x,0)\in \sigma ({(T_1^{\ast })}^{-1}(({\gamma }_1+\alpha ,1]))\cap \mathrm{\cdots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n+\alpha ,1])).$$

From the assumption on $X\times \{0\}$, there is some path connected open set $V^{\ast }$ such that

$$(x,0)\in V^{\ast }\subseteq \sigma ({(T_1^{\ast })}^{-1}(({\gamma }_1+\alpha ,1]))\cap \mathrm{\cdots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n+\alpha ,1])).$$

Since $\sigma$ is continuous, then ${\sigma }^{-1}(V^{\ast })$ is an open neighbourhood of $(x,\alpha )$, then so is

$$U^{\ast }={\sigma }^{-1}(V^{\ast })\cap {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1]).$$

We prove that for every two points $(x_1,{\alpha }_1),(x_2,{\alpha }_2)\in U^{\ast }$, there is a path in ${(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1])$ connecting these two points. Their images $(x_1,0),(x_2,0)$ under $\sigma$ are in $V^{\ast }$, therefore there is a path $p^{\ast }$ connecting $(x_1,0)$ to $(x_2,0)$ which is included in $\sigma ({(T_1^{\ast })}^{-1}(({\gamma }_1+\alpha ,1]))\cap \mathrm{\cdots }\cap \sigma ({(T_n^{\ast })}^{-1}(({\gamma }_n+\alpha ,1]))$. This means that for every point $(y,0)$ in $p^{\ast }$ we have that for every $i=1,\mathrm{\dots },n$, $T_i(y)>{\gamma }_i+\alpha$ which implies that

$$(y,\alpha )\in {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1]).$$

It follows that the path ${}_{\alpha }{}^{}p_{}^{\ast }$ which connects $(x_1,\alpha )$ to $(x_2,\alpha )$ is such that

$$Im({}_{\alpha }{}^{}p_{}^{\ast })\subset {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1]).$$

Since

$$(x_1,\alpha ),(x_2,\alpha ),(x_1,{\alpha }_1),(x_2,{\alpha }_2)\in {(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1]),$$

we have from Lemma 3.7 the existence of paths $p_1$ and $p_2$ inside

$${(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1])$$

which connect respectively $(x_1,{\alpha }_1)$ to $(x_1,\alpha )$, and $(x_2,\alpha )$ to $(x_2,{\alpha }_2)$. The composite $p_1\ast {}_{\alpha }{}^{}p_{}^{\ast }\ast p_2$ is a path inside

$${(T_1^{\ast })}^{-1}(({\gamma }_1,1])\cap \mathrm{\cdots }\cap {(T_n^{\ast })}^{-1}(({\gamma }_n,1])\cap P_2^{-1}((\delta ,1])$$

which connects $(x_1,{\alpha }_1)$ to $(x_2,{\alpha }_2)$.

Conversely assume that $X\times J$ is l.p.c. Let $U^{\ast }$ be any open in $X\times \{0\}$ containing any $(x,0)$. Consider ${\sigma }^{-1}(U^{\ast })$ which is open in $X\times J$ and contains $(x,0)$. Since $X\times J$ is l.p.c., there exists an open neighbourhood $V$ of $(x,0)$ such that $V\subseteq {\sigma }^{-1}(U^{\ast })$ and satisfies the property that any two points in $V$ are connected by a path which is included in $V$. Decompose $V$ as $V=\underset{i\in ℐ}{\cup }{C}_i$ where each $C_i=\underset{j\in {𝒥}_i}{\cap }{B}_{i,j}$ with ${𝒥}_i$ finite and each $B_{i,j}$ being a sub basis member. There is some $i\in ℐ$ such that $(x,0)\in C_i=\underset{j\in {𝒥}_i}{\cap }{B}_{i,j}$. If for some $j\in {𝒥}_i$, $B_{i,j}=P_2^{-1}(({\gamma }_{i,j},1])$, then , therefore $B_{i,j}=X\times J$. It follows that, if for all $j\in {𝒥}_i$, $B_{i,j}=P_2^{-1}(({\gamma }_{i,j},1])$, then $C_i=X\times J$, consequently $V=X\times J={\sigma }^{-1}(U^{\ast })$. In this case we prove that $U^{\ast }$ has the property that any two points in $U^{\ast }$ are connected by a path that is included in $U^{\ast }$. Indeed, let $(x_1,0),(x_2,0)\in U^{\ast }$. Their preimages $(x_1,0),(x_2,0)\in V$ can be connected by a path $p$ in $V={\sigma }^{-1}(U^{\ast })$. Then $\sigma (p)$ is a path in $U^{\ast }$ connecting $(x_1,0)$ to $(x_2,0)$. Now assume that not for all $j\in {𝒥}_i$, $B_{i,j}=P_2^{-1}(({\gamma }_{i,j},1])$. It follows that there is a non-empty subset ${𝒥}_i^{\prime }$ of ${𝒥}_i$ such that for every $j\in {𝒥}_i^{\prime }$, $B_{i,j}={(T_j^{\ast })}^{-1}(({\gamma }_j,1])$ and for all $j\notin {𝒥}_i^{\prime }$, $B_{i,j}=P_2^{-1}(({\gamma }_j,1])=X\times J$. In such circumstances, $C_i=\underset{j\in {𝒥}_i^{\prime }}{\cap }{(T_j^{\ast })}^{-1}(({\gamma }_j,1])$. From Lemma 3.3 and Lemma 3.9 we have that $\sigma (C_i)$ is an open neighbourhood of $(x,0)$ in $X\times \{0\}$. Let $(x_1,0),(x_2,0)\in \sigma (C_i)$ be arbitrary. Since

$$\sigma (C_i)=\sigma (\underset{j\in {𝒥}_i^{\prime }}{\cap }{(T_j^{\ast })}^{-1}(({\gamma }_j,1]))=\underset{j\in {𝒥}_i^{\prime }}{\cap }\sigma ({(T_j^{\ast })}^{-1}({\gamma }_j,1]))=\underset{j\in {𝒥}_i^{\prime }}{\cap }{(T_j^{\ast })}^{-1}(({\gamma }_j,1]))\cap (X\times \{0\}),$$

it follows that $(x_1,0),(x_2,0)\in \underset{j\in {𝒥}_i^{\prime }}{\cap }{(T_j^{\ast })}^{-1}(({\gamma }_j,1]))\subseteq V$. From the assumption on $V$, there is a path $p$ in $V$ connecting $(x_1,0)$ to $(x_2,0)$. Since $V\subseteq {\sigma }^{-1}(U^{\ast })$, it follows that $\sigma (p)$ is a path in $U^{\ast }$ connecting $(x_1,0)$ to $(x_2,0)$ proving that $X\times \{0\}$ is locally path connected.
(ii) Assume that $X\times \{0\}$ is path-connected, and let $(x,\alpha ),(y,\beta )\in X\times J$ be arbitrary. There is a path $p$ connecting $(x,\alpha )$ to $(x,0)$, and a path $q$ connecting $(y,\beta )$ to $(y,0)$. Also, from the assumption, there is a path $r$ connecting $(x,0)$ to $(y,0)$. Now the path $p\ast r\ast q^{-1}$ connects $(x,\alpha )$ to $(y,\beta )$ proving that $X\times J$ is path connected. Conversely, for every $(x,0),(y,0)\in X\times \{0\}$, and every path $p$ in $X\times J$ which connects $(x,0)$ to $(y,0)$, $\sigma (p)$ is a path in $X\times \{0\}$ which connects $(x,0)$ to $(y,0)$ proving that $X\times \{0\}$ is path connected. This completes the proof.

Lemma 3.13.

Let $\gamma :I\to X\times J$ be a path in $X\times J$, and $t\in I$ a fixed element. Let $I\times (X\times J)$ be the product of spaces equipped with the product topology. The map ${\gamma }_t:I\to I\times (X\times J)$ defined by ${\gamma }_t(s)=(t,\gamma (s))$ is continuous, and therefore a path in $I\times (X\times J)$. This path is in fact a pair $(c_t,\gamma )$ of paths with $c_t$ the constant path at $t$ in $I$, and $\gamma$ the given path in $X\times J$.

Proof 3.14.

Since $I\times (X\times J)$ is equipped with the product topology, the continuity of ${\gamma }_t$ follows from the fact that its compositions with projections in the first and in the second coordinate, give respectively, the constant map at $t$, and the map $\gamma$ which are both continuous. The second statement is based on the standard fact that paths in a product of two spaces equipped with the product topology correspond to pairs of paths by taking compositions with the respective projections.

In the rest of the paper, wherever it appears, $H$ will be the deformation retraction defined in Theorem 2.14.

Lemma 3.15.

Let $\gamma :I\to (X\times J)$ be a path with image inside $X\times \{0\}$ and $t\in I$ arbitrary, then $H\circ {\gamma }_t=\gamma$.

Proof 3.16.

Let for any $s\in I$, $(x,0)=\gamma (s)$. Then,

$$(H\circ {\gamma }_t)(s)=H(t,\gamma (s))=H(t,(x,0))=(x,0)=\gamma (s),$$

which shows that $H\circ {\gamma }_t=\gamma$.

Lemma 3.17.

Let $\gamma :I\to (X\times J)$ be any path, then $H\circ {\gamma }_1$ is a path inside $X\times \{0\}$.

Proof 3.18.

Let for any $s\in I$, $(x,\alpha )=\gamma (s)$. Then,

$$(H\circ {\gamma }_1)(s)=H(1,\gamma (s))=H(1,(x,\alpha ))=(x,(1-1)\alpha )=(x,0),$$

which shows that $\text{Im}(H\circ {\gamma }_1)\subseteq X\times \{0\}$.

Lemma 3.19.

Let $\gamma :I\to (X\times J)$ be any path, then $H\circ {\gamma }_0=\gamma$.

Proof 3.20.

Let for any $s\in I$, $(x,\alpha )=\gamma (s)$. Then,

$$(H\circ {\gamma }_0)(s)=H(0,\gamma (s))=H(0,(x,\alpha ))=(x,(1-0)\alpha )=(x,\alpha )=\gamma (s),$$

which shows that $H\circ {\gamma }_0=\gamma$.

Lemma 3.21.

For every path $\gamma :I\to X\times J$ and every $s,t\in I$ with $s\ne t$, the paths $H\circ {\gamma }_s$ and $H\circ {\gamma }_t$ are freely homotopic in $X\times J$.

Proof 3.22.

We let first $\kappa (s,t):I\to I$ be the map $\kappa (s,t)(x)=(t-s)x+s$ which is continuous and satisfies $\kappa (s,t)(0)=s$ and $\kappa (s,t)(1)=t$, hence it is a path in $I$ from $s$ to $t$. Now we define

$${\chi }_{\gamma ,\kappa (s,t)}:I\times I\to X\times J$$

by setting

$${\chi }_{\gamma ,\kappa (s,t)}(\eta ,x)=H(\kappa (s,t)(x),\gamma (\eta )).$$

For $x=0$ we have that

$${\chi }_{\gamma ,\kappa (s,t)}(\eta ,0)=H(s,\gamma (\eta ))=(H\circ {\gamma }_s)(\eta ),$$

and for $x=1$ we have that

$${\chi }_{\gamma ,\kappa (s,t)}(\eta ,1)=H(t,\gamma (\eta ))=(H\circ {\gamma }_t)(\eta ).$$

The map ${\chi }_{\gamma ,\kappa (s,t)}$ is continuous. Indeed, ${\chi }_{\gamma ,\kappa (s,t)}$ factors through $H$ as ${\chi }_{\gamma ,\kappa (s,t)}=H\circ (\kappa (s,t),\gamma )\circ j$ where $(\kappa (s,t),\gamma ):I\times I\to I\times (X\times J)$ is defined by $(x,\eta )\mapsto (\kappa (s,t)(x),\gamma (\eta ))$ and $j:I\times I\to I\times I$ is defined by $j(\eta ,x)=(x,\eta )$. The continuity of ${\chi }_{\gamma ,\kappa (s,t)}$ follows from that of $j$ and that of $(\kappa (s,t),\gamma )$. The map $j$ is obviously continuous, and $(\kappa (s,t),\gamma )$ is continuous too since its composite with the projection in the first coordinate gives $\kappa (s,t)$ and its composite with the projection in the second coordinate gives $\gamma$ which are both continuous.

Remark 3.23.

Let $\gamma :I\to X\times J$ be a path with $\iota (\gamma )=\gamma (0)=(y,\alpha )$ and $\tau (\gamma )=\gamma (1)=(z,\beta )$. Let $s,t\in I$ and let ${\chi }_{\gamma ,\kappa (s,t)}$ be the free homotopy of Lemma 3.21 from $H\circ {\gamma }_s$ to $H\circ {\gamma }_t$. The restriction of the homotopy ${\chi }_{\gamma ,\kappa (s,t)}$ on $\{0\}\times I$ induces a path $p$ given by

$$p(x)={\chi }_{\gamma ,\kappa (s,t)}(0,x)=H(\kappa (s,t)(x),\gamma (0)).$$

We note that

$$p(0)=(H\circ {\gamma }_s)(0)=H(s,\gamma (0))=(y,(1-s)\alpha )$$

and

$$p(1)=(H\circ {\gamma }_t)(0)=H(t,\gamma (0))=(y,(1-t)\alpha ).$$

Similarly, by restricting ${\chi }_{\gamma ,\kappa (s,t)}$ on $\{1\}\times I$ we get a path $q$ given by

$$q(x)={\chi }_{\gamma ,\kappa (s,t)}(1,x)=H(\kappa (s,t)(x),\gamma (1)),$$

which runs from

$$q(0)=(H\circ {\gamma }_s)(1)=H(s,\gamma (1))=(z,(1-s)\beta )$$

to

$$q(1)=(H\circ {\gamma }_t)(1)=H(t,\gamma (1))=(z,(1-t)\beta ).$$

We observe that $p^{-1}\ast (H\circ {\gamma }_s)\ast q$ and $H\circ {\gamma }_t$ are now parallel paths with

$$\iota (p^{-1}\ast (H\circ {\gamma }_s)\ast q)=\iota (p^{-1})=(y,(1-t)\alpha )=(H\circ {\gamma }_t)(0),$$

and

$$\tau (p^{-1}\ast (H\circ {\gamma }_s)\ast q)=\tau (q)=(z,(1-t)\beta )=(H\circ {\gamma }_t)(1).$$

Also we note that the restriction of ${\chi }_{\gamma ,\kappa (s,t)}$ on $I\times \{0\}$ gives $H\circ {\gamma }_s$ since

$${\chi }_{\gamma ,\kappa (s,t)}(\eta ,0)=H(\kappa (s,t)(0),\gamma (\eta ))=H(s,\gamma (\eta ))=(H\circ {\gamma }_s)(\eta ).$$

In a similar fashion one can verify that the restriction of ${\chi }_{\gamma ,\kappa (s,t)}$ on $I\times \{1\}$ gives $H\circ {\gamma }_t$.

Finally, we pay a special attention to the case when $\gamma$ is the constant path $c_{(y,\alpha )}$ at some $(y,\alpha )$, $s=0$ and $t\in I$ arbitrary. In this case the above path $p$ starts at $p(0)=H(0,(y,\alpha ))=(y,\alpha )$ and ends at $p(1)=H(t,(y,\alpha ))=(y,(1-t)\alpha )$. This implies that for every two points $(y,\alpha ),(y,\beta )\in X\times J$, there is always a path in $X\times J$ connecting $(y,\alpha )$ to $(y,\beta )$ consisting of points $(y,\delta )$ with $\delta$ between $\alpha$ and $\beta$. Indeed, assuming that one of $\alpha$ or $\beta$ is non zero, for instance $\alpha \ne 0$, we can take $s=0$ and $t=1-{\displaystyle \frac{\beta }{\alpha }}$ and that will do.

Lemma 3.24.

The paths $p^{-1}\ast (H\circ {\gamma }_s)\ast q$ and $H\circ {\gamma }_t$ are homotopic relative to their end points.

Proof 3.25.

The map ${\chi }_{\gamma ,\kappa (s,t)}$ is null homotopic since $I\times I$ is contractible. On the other hand, $I\times I$ is homeomorphic with $D^2$ and its boundary is homeomorphic with $S^1$. Since the inclusion map $\iota :S^1\to D^2$ is null homotopic, it follows that the restriction of ${\chi }_{\gamma ,\kappa (s,t)}$ on the boundary of $I\times I$ is null homotopic too. But from Remark 3.23 the image of this restriction is the path $p^{-1}\ast (H\circ {\gamma }_s)\ast q\ast {(H\circ {\gamma }_t)}^{-1}$, hence $p^{-1}\ast (H\circ {\gamma }_s)\ast q\ast {(H\circ {\gamma }_t)}^{-1}\sim c_{(y,(1-t)\alpha )}$. This implies that

$p^{-1}\ast (H\circ {\gamma }_s)\ast q$ $\sim p^{-1}\ast (H\circ {\gamma }_s)\ast q\ast c_{(z,(1-t)\beta )}$

$\sim p^{-1}\ast (H\circ {\gamma }_s)\ast q\ast {(H\circ {\gamma }_t)}^{-1}\ast (H\circ {\gamma }_t)$

$\sim c_{(y,(1-t)\alpha )}\ast (H\circ {\gamma }_t)$

$\sim H\circ {\gamma }_t,$

which proves the claim.

Lemma 3.26.

For every path $\gamma :I\to (X\times J)$ and every $t\in I$, ${(H\circ {\gamma }_t)}^{-1}=H\circ {\gamma }_t^{-1}$.

Proof 3.27.

We wee that for every $u\in I$,

${(H\circ {\gamma }_t)}^{-1}(u)$ $=(H\circ {\gamma }_t)(1-u)$

$=H(t,\gamma (1-u))$

$=H(t,{\gamma }^{-1}(u))$

$=(H\circ {\gamma }_t^{-1})(u),$

proving the claim.

Lemma 3.28.

For every $n$ paths ${\gamma }_1,\mathrm{\dots },{\gamma }_n$ in $X\times J$ such that the concatenation ${\gamma }_1\ast \mathrm{\cdots }\ast {\gamma }_n$ exists, and for every fixed $t\in I$, the concatenation $(H\circ {({\gamma }_1)}_t)\ast \mathrm{\cdots }\ast (H\circ {({\gamma }_n)}_t)$ exists, and $(H\circ {({\gamma }_1)}_t)\ast \mathrm{\cdots }\ast (H\circ {({\gamma }_n)}_t)=H\circ {({\gamma }_1\ast \mathrm{\cdots }\ast {\gamma }_n)}_t$.

Proof 3.29.

Indeed, since for every $1\le i\le n-1$, ${\gamma }_i(1)={\gamma }_{i+1}(0)$, then

$$(H\circ {({\gamma }_i)}_t)(1)=H(t,{\gamma }_i(1))=H(t,{\gamma }_{i+1}(0))=(H\circ {({\gamma }_{i+1})}_t)(0),$$

which proves the first part of the lemma. The proof of the second part will be done by induction on $n\ge 2$. When $n=2$, for every $s\in [0,1/2]$,

$(H\circ {({\gamma }_1\ast {\gamma }_2)}_t)(s)$ $=H(t,({\gamma }_1\ast {\gamma }_2)(s))=H(t,{\gamma }_1(2s))$

$=((H\circ {({\gamma }_1)}_t)\ast (H\circ {({\gamma }_2)}_t))(s),$

and similarly, for $s\in [1/2,1]$,

$(H\circ {({\gamma }_1\ast {\gamma }_2)}_t)(s)$ $=H(t,({\gamma }_1\ast {\gamma }_2)(s))=H(t,{\gamma }_2(2s-1))$

$=((H\circ {({\gamma }_1)}_t)\ast (H\circ {({\gamma }_2)}_t))(s).$

By comparing we see that $H\circ {({\gamma }_1\ast {\gamma }_2)}_t=(H\circ {({\gamma }_1)}_t)\ast (H\circ {({\gamma }_2)}_t)$. For the inductive step, considering that ${\gamma }_1\ast \mathrm{\cdots }\ast {\gamma }_{n-1}\ast {\gamma }_n=({\gamma }_1\ast \mathrm{\cdots }\ast {\gamma }_{n-1})\ast {\gamma }_n$ by definition, we have that

$H\circ {({\gamma }_1\ast \mathrm{\cdots }\ast {\gamma }_{n-1}\ast {\gamma }_n)}_t$ $=(H\circ {({\gamma }_1\ast \mathrm{\cdots }\ast {\gamma }_{n-1})}_t)\ast (H\circ {({\gamma }_n)}_t)$ $(\text{by the base step})$

$=(H\circ {({\gamma }_1)}_t)\ast \mathrm{\cdots }\ast (H\circ {({\gamma }_{n-1})}_t)\ast (H\circ {({\gamma }_n)}_t)$ $(\text{by the assumption,})$

which proves the second part of the lemma.

From this moment and on we will assume that $(X,{\iota }_X(𝒯))$ is p.c and l.p.c. to ensure that we can define the topological space ${\pi }^l(X\times J)$.

Theorem 3.30.

The space ${\pi }^l(X\times J)$ deformation retracts to its subspace ${\pi }^l(X\times \{0\})$.

Proof 3.31.

We define

$$H^l:I\times {\pi }^l(X\times J)\to {\pi }^l(X\times J)\text{ by }(t,[\gamma ])\mapsto [H\circ {\gamma }_t].$$

This map is well defined for if $\gamma \sim \delta$ are homotopic paths in $X\times J$ (relative to end points) and $t\in I$ fixed, then

$$P_2\circ {\gamma }_t=\gamma \sim \delta =P_2\circ {\delta }_t.$$

But for both, $\gamma$ and $\delta$,

$$P_1({\gamma }_t)=c_t=P_1({\delta }_t),$$

where $c_t$ is the constant path at $t$ in $I$, therefore we have a homotopy of pairs

$${\gamma }_t=(c_t,\gamma )\sim (c_t,\delta )={\delta }_t$$

in $I\times (X\times J)$. The continuity of $H$ now implies that $H\circ {\gamma }_t\sim H\circ {\delta }_t$, and so $[H\circ {\gamma }_t]=[H\circ {\delta }_t]$.

Before we prove the continuity of $H^l$ we define an open cover ${𝒰}^{\prime }$ of $X\times J$ in terms of any given open cover $𝒰$ of $X\times J$ and of $H$ in the following way. Since $H$ is continuous, for every $V\in 𝒰$, $H^{-1}(V)$ is open in $I\times (X\times J)$, therefore for some index set ${\Im }_V$, $H^{-1}(V)=\underset{i\in {\Im }_V}{\cup }{V}_i^{(1)}\times V_i^{(2)}$ where $V_i^{(1)}$ is open in $I$ and $V_i^{(2)}$ is open in $X\times J$. The family ${𝒰}^{\prime }$ consisting of all $V_i^{(2)}$ with $V$ ranging in $𝒰$ and $i\in {\Im }_V$ is an open cover for $X\times J$. Indeed, for every $(x,\alpha )\in X\times J$, there is $V\in 𝒰$ such that $(x,\alpha )\in V$. It follows that $(0,(x,\alpha ))\in H^{-1}(V)=\underset{i\in {\Im }_V}{\cup }{V}_i^{(1)}\times V_i^{(2)}$ therefore for some $i\in {\Im }_V$, $(0,(x,\alpha ))\in V_i^{(1)}\times V_i^{(2)}$, hence $(x,\alpha )\in V_i^{(2)}$, where from the definition, $V_i^{(2)}\in {𝒰}^{\prime }$.

Now we prove the continuity of $H^l$. Let $(t,[\gamma ])\in I\times {\pi }^l(X\times J)$ and assume that $N([H\circ {\gamma }_t],𝒰,V,W)$ is an arbitrary open neighbourhood of $[H\circ {\gamma }_t]$. If $\iota (\gamma )=(x,\alpha )$ and $\tau (\gamma )=(y,\beta )$, then $\iota (H\circ {\gamma }_t)=(x,(1-t)\alpha )$ and $\tau (H\circ {\gamma }_t)=(y,(1-t)\beta )$. So $V$ is an open neighbourhood of $(x,(1-t)\alpha )$ and $W$ an open neighbourhood of $(y,(1-t)\beta )$. It follows that $(t,(x,\alpha ))\in H^{-1}(V)=\underset{i\in {\Im }_V}{\cup }{V}_i^{(1)}\times V_i^{(2)}$, therefore for some $i\in {\Im }_V$, $t\in V_i^{(1)}$ and $(x,\alpha )\in V_i^{(2)}$. Since $V_i^{(2)}$ is an open neighbourhood of $(x,\alpha )$, we rewrite it for simplicity as $V_{(x,\alpha )}$. It is also clear from the definition of ${𝒰}^{\prime }$ that $V_{(x,\alpha )}\in {𝒰}^{\prime }$. In the same way we find an open neighbourhood $W_j^{(1)}$ of $t$ in $I$ and an open neighbourhood $W_{(y,\beta )}$ of $(y,\beta )$ in $X\times J$ which is a member of ${𝒰}^{\prime }$. Since $V_i^{(1)}\cap W_j^{(1)}$ is an open subset of $I$ containing $t$, there is an open "interval" ${\Im }_t$ such that $t\in {\Im }_t$ and ${\Im }_t\subset V_i^{(1)}\cap W_j^{(1)}$. It is now clear that an open neighbourhood of $(t,[\gamma ])$ in $I\times {\pi }^l(X\times J)$ is the product ${\Im }_t\times N([\gamma ],{𝒰}^{\prime },V_{(x,\alpha )},W_{(y,\beta )})$. Next we prove that for every $(s,[\delta ])\in {\Im }_t\times N([\gamma ],{𝒰}^{\prime },V_{(x,\alpha )},W_{(y,\beta )})$ we have that $H^l((s,[\delta ]))=[H\circ {\delta }_s]\in N([H\circ {\gamma }_t],𝒰,V,W)$. From the definition of $N([\gamma ],{𝒰}^{\prime },V_{(x,\alpha )},W_{(y,\beta )})$,

$$\delta \sim \lambda \ast \mu \ast \gamma \ast {\mu }^{\prime }\ast \omega$$ (5)

where $\lambda :I\to V_{(x,\alpha )}$ with $\tau (\lambda )=(x,\alpha )$, $\mu \in {\pi }_1({𝒰}^{\prime },(x,\alpha ))$, ${\mu }^{\prime }\in {\pi }_1({𝒰}^{\prime },(y,\beta ))$ and $\omega :I\to W_{(y,\beta )}$ with $\iota (\omega )=(y,\beta )$. It follows from the continuity of $H$ and from Lemma 3.28 that

$$H\circ {\delta }_s\sim (H\circ {\lambda }_s)\ast (H\circ {\mu }_s)\ast (H\circ {\gamma }_s)\ast (H\circ {\mu }_s^{\prime })\ast (H\circ {\omega }_s).$$ (6)

Since $s\in {\Im }_t$, and since for every $u\in I$, $\lambda (u)\in V_{(x,\alpha )}$, it follows that

$$(H\circ {\lambda }_s)(u)=H((s,\lambda (u)))\subseteq H(V_i^{(1)}\times V_{(x,\alpha )})\subseteq H(H^{-1}(V))=V,$$ (7)

hence $H\circ {\lambda }_s$ is a path with $Im(H\circ {\lambda }_s)\subseteq V$. Also we see that $\tau (H\circ {\lambda }_s)=(x,(1-s)\alpha )$ since $\lambda (1)=(x,\alpha )$. By a similar argument we see that $H\circ {\omega }_s$ is a path with $\text{Im}(H\circ {\omega }_s)\subseteq W$ and $\iota (H\circ {\omega }_s)=(y,(1-s)\beta )$. Further, since $\mu$ is a finite concatenation of paths

$$\mu =\prod _{𝑖}{u}_i\ast v_i\ast u_i^{-1}$$

where each $u_i$ is a path starting at $(x,\alpha )$ and ending at some $(x_i,{\alpha }_i)$ and $v_i$ is a loop at $(x_i,{\alpha }_i)$ which lies entirely inside some $U_i^{\prime }\in {𝒰}^{\prime }$, it follows that

$H\circ {\mu }_s$ $={\displaystyle \prod }_{𝑖}(H\circ {(u_i)}_s)\ast (H\circ {(v_i)}_s)\ast (H\circ {(u_i^{-1})}_s)$ (8)

$={\displaystyle \prod }_{𝑖}(H\circ {(u_i)}_s)\ast (H\circ {(v_i)}_s)\ast {(H\circ {(u_i)}_s)}^{-1}$ $(\text{by lemma }\text{<a href="\#S3.Thmtheorem26" title="Lemma 3.26. ‣ 3. The groupoid approach ‣ A groupoid approach to the study of fuzzy topological spaces" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">3.26</span></a>})$

where

$$(H\circ {(u_i)}_s)(0)=H((s,(x,\alpha )))=(x,(1-s)\alpha ),$$

and that

$$(H\circ {(u_i)}_s)(1)=H((s,(x_i,{\alpha }_i)))=(x_i,(1-s){\alpha }_i).$$

Since $U_i^{\prime }\in {𝒰}^{\prime }$, there has been some $U_i\in 𝒰$ and an open subset $A\subseteq I$ such that $A\times U_i^{\prime }$ takes part in the decomposition of $H^{-1}(U_i)$ as a union of direct products. If $s\in A$, then for every $r\in I$, since $v_i(r)\in U_i^{\prime }$, it follows that $(s,v_i(r))\in H^{-1}(U_i)$, consequently we have that

$$(H\circ {(v_i)}_s)(r)=H((s,v_i(r)))\in H(H^{-1}(U_i))\subseteq U_i.$$

So in this case the loop $H\circ {(v_i)}_s$ is included in $U_i\in 𝒰$. Now we deal with the case when $s\notin A$. For any $s_i\in A$, we have from the above that $H\circ {(v_i)}_{s_i}$ is included in $U_i$. Let $\kappa (s_i,s):I\to I$ be the path in $I$ of Lemma 3.21 with $\kappa (s_i,s)(0)=s_i$ and $\kappa (s_i,s)(1)=s$ which defines a free homotopy ${\chi }_{v_i,\kappa (s_i,s)}$ from $H\circ {(v_i)}_{s_i}$ to $H\circ {(v_i)}_s$ by the rule

$${\chi }_{v_i,\kappa (s_i,s)}(r,p)=H((\kappa (s_i,s)(p),v_i(r))).$$

For $p=0$ we have that

$${\chi }_{v_i,\kappa (s_i,s)}(r,0)=H((s_i,v_i(r)))=(H\circ {(v_i)}_{s_i})(r),$$

and for $p=1$ we have that

$${\chi }_{v_i,\kappa (s_i,s)}(r,1)=H((s,v_i(r)))=(H\circ {(v_i)}_s)(r).$$

We observe that $H\circ {(v_i)}_{s_i}$ is a loop at

$$(H\circ {(v_i)}_{s_i})(0)=H((s_i,v_i(0)))=H((s_i,(x_i,{\alpha }_i)))=(x_i,(1-s_i){\alpha }_i),$$

and most importantly that,

$$(H\circ {(v_i)}_{s_i})(r)=H((s_i,v_i(r)))\in H(H^{-1}(U_i))\subseteq U_i,$$

since $s_i\in A$. The restriction of ${\chi }_{v_i,\kappa (s_i,s)}$ on $\{0\}\times I$ gives a path $p_i$

But from Lemma 3.24 we have that

$$H\circ {(v_i)}_s\sim p_i^{-1}\ast (H\circ {(v_i)}_{s_i})\ast p_i.$$

It follows that we can now replace in (8) each component

$$(H\circ {(u_i)}_s)\ast (H\circ {(v_i)}_s)\ast {(H\circ {(u_i)}_s)}^{-1}$$

whenever $s\notin A$ by

$$((H\circ {(u_i)}_s)\ast p_i^{-1})\ast (H\circ {(v_i)}_{s_i})\ast {((H\circ {(u_i)}_s)\ast p_i^{-1})}^{-1}$$

without changing the homotopy class of $H\circ {\mu }_s$. In conclusion, $H\circ {\mu }_s$ is homotopic with a loop in ${\pi }_1(𝒰,(x,(1-s)\alpha ))$. By a similar argument, we see that $H\circ {\mu }_s^{\prime }$ is homotopic with a loop in ${\pi }_1(𝒰,(y,(1-s)\beta ))$. What we proved so far, together with (6), imply that $[H\circ {\delta }_s]\in N([H\circ {\gamma }_s],𝒰,V,W)$. A particular case of this is when $\delta =\gamma$, and $\lambda ,\mu ,{\mu }^{\prime },\omega$ are trivial. Then from (7) we would have for every $\xi \in {\Im }_t$ that

$$(x,(1-\xi )\alpha )=H((\xi ,(x,\alpha )))=H((\xi ,c_{(x,\alpha )}(u)))=(H\circ {(c_{(x,\alpha )})}_{\xi })(u)\subseteq V.$$ (9)

We prove now that $[H\circ {\gamma }_t]\in N([H\circ {\gamma }_s],𝒰,V,W)$ which together with Lemma 2.3 of [6] imply that $[H\circ {\delta }_s]\in N([H\circ {\gamma }_t],𝒰,V,W)$ as desired. We will utilize again Remark 3.23 in the following form. Let $\kappa (s,t):I\to I$ be the map of Lemma 3.21 such that $\kappa (s,t)(0)=s$ and $\kappa (s,t)(1)=t$ and let ${\chi }_{\gamma ,\kappa (s,t)}$ be the corresponding free homotopy from $H\circ {\gamma }_s$ to $H\circ {\gamma }_t$. The restriction of ${\chi }_{\gamma ,\kappa (s,t)}$ on $\{0\}\times I$ gives a path

$$p(\xi )={\chi }_{\gamma ,\kappa (s,t)}(0,\xi )=H((\kappa (s,t)(\xi ),\gamma (0)))$$

with

$$p(0)=H((s,\gamma (0)))=(x,(1-s)\alpha )\text{ and }p(1)=H((t,\gamma (0)))=(x,(1-t)\alpha ),$$

which from (9) is included entirely in $V$. By restricting now ${\chi }_{\gamma ,\kappa (s,t)}$ on $\{1\}\times I$ we get a path

$$q(\xi )={\chi }_{\gamma ,\kappa (s,t)}(1,\xi )=H((\kappa (s,t)(\xi ),\gamma (1))),$$

with

$$q(0)=H(s,\gamma (1))=(y,(1-s)\beta )\text{ and }q(1)=H((t,\gamma (1)))=(y,(1-t)\beta ).$$

By a similar argument with the one used to prove that $\text{Im}p\subset U$, one can prove that $\text{Im}q\subset W$. It follows that $[p^{-1}\ast (H\circ {\gamma }_s)\circ q]\in N([H\circ {\gamma }_s],𝒰,V,W)$. Now Lemma 3.24 implies that

$$H\circ {\gamma }_t\sim p^{-1}\ast (H\circ {\gamma }_s)\circ q,$$

which proves that $[H\circ {\gamma }_t]\in N([H\circ {\gamma }_s],𝒰,V,W)$. Finally, we need to check the rest of the conditions for $H^l$ to be a deformation retraction. For every $[\gamma ]\in H^l(X\times J)$ we have from Lemma 3.19 that

$$H^l((0,[\gamma ]))=[H\circ {\gamma }_0]=[\gamma ].$$

For every $[\gamma ]\in H^l(X\times J)$ we have from Lemma 3.17 that

$$H^l((1,[\gamma ]))\in H^l(X\times \{0\}).$$

For every $t\in I$ and every path $\gamma$ in $X\times \{0\}$ we have from Lemma 3.15 that

$$H^l((t,[\gamma ]))=[\gamma ],$$

which concludes the proof.

Corollary 3.32.

There is a continuous map ${\pi }^l\sigma :{\pi }^l(X\times J,\iota (𝒯))\to {\pi }^l(X\times \{0\},\iota (𝒯))$.

Proof 3.33.

The existence of ${\pi }^l\sigma$ follows directly from Proposition 2.6 of [6] and is defined by ${\pi }^l\sigma ([\alpha ])=[\sigma (\alpha )]$ for every homotopy class of a path $\alpha$ in $X\times J$. Also ${\pi }^l\sigma$ can be realized as the restriction of $H^l$ on $\{1\}\times {\pi }^l(X\times J)$ since $H^l((1,[\alpha ]))=[H\circ {\alpha }_1]$ and $(H\circ {\alpha }_1)(s)=H((1,\alpha (s)))=\sigma (\alpha )(s)$.

4. Fuzzy complements in topological terms

As we mentioned in section 2 the main goal of this paper is interpreting in topological terms the complement of a fuzzy subset. It turns out that the groupoid $\pi ({\pi }^l(X\times J))$ has the necessary information encoded in it to make this interpretation possible. Theorem 4.16 explained in simple terms, states that every fuzzy subset $F$ and its complement $1-F$, give rise to a pair of functors $[F],[1-F]$ in the category Grpd of groupoids which are related by the equality $[1-F]=\pi (𝜾)\circ [F]$ where $\pi (𝜾):\pi ({\pi }^l(X\times J))\to \pi ({\pi }^l(X\times J)$ is the natural extension to the homotopy classes of paths of the map $𝜾:{\pi }^l(X\times J)\to {\pi }^l(X\times J)$ which maps every $[\alpha ]$ to $[{\alpha }^{-1}]$. Therefore complementing $F$ in ${[0,1]}^X$, is represented in Grpd by applying $\pi (𝜾)$ to $[F]$ where $\pi (𝜾)$ maps points to their inverses. We emphasize here that we do not assume any topology defined on $\pi ({\pi }^l(X\times J))$. It is just the fundamental groupoid of the topological space ${\pi }^l(X\times J)$.

Let $\gamma$ be a path in $X\times J$, $s,t\in I$ and let ${\chi }_{\gamma ,\kappa (s,t)}$ be the free homotopy of Lemma 3.21 which transforms $H\circ {\gamma }_s$ to $H\circ {\gamma }_t$. For every we let ${\gamma }^{\prime }:I\to X\times J$ be the path given by ${\gamma }^{\prime }(\eta )=\gamma (a+\eta (b-a))$. In the following lemma we will describe the free homotopy of Lemma 3.21 which transforms $H\circ {\gamma }_s^{\prime }$ to $H\circ {\gamma }_t^{\prime }$ in terms of ${\chi }_{\gamma ,\kappa (s,t)}$. Our interest lies in studying the neighbourhoods of the homotopy classes of paths of Remark 2.12 arising from the above homotopy. For this we need the following. For every fixed $\eta \in I$ we let ${\mathrm{\ell }}_{\eta }:I\to I\times I$ be the continuous map $x\mapsto (\eta ,x)$. Its composite ${\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }$ is continuous and is given by the rule $({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta })(x)=H(\kappa (s,t)(x),\gamma (\eta ))=(H\circ {\gamma }_{\kappa (s,t)(x)})(\eta )$.

Lemma 4.1.

Let $\gamma$ be a path in $X\times J$, $s,t\in I$ and let ${\chi }_{\gamma ,\kappa (s,t)}$ be the free homotopy of Lemma 3.21 which transforms $H\circ {\gamma }_s$ to $H\circ {\gamma }_t$. For every , the free homotopy induced by ${\chi }_{\gamma ,\kappa (s,t)}$ from the restriction of $H\circ {\gamma }_s$ in $[a,b]$ to the restriction of $H\circ {\gamma }_t$ in $[a,b]$ coincides with the free homotopy from $H\circ {\gamma }_s^{\prime }$ to $H\circ {\gamma }_t^{\prime }$. The path of Remark 3.23 arising from this homotopy which connects $(H\circ {\gamma }_s)(a)$ to $(H\circ {\gamma }_t)(a)$ is given by $({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_a)(x)$, and the other path which connects $(H\circ {\gamma }_s)(b)$ to $(H\circ {\gamma }_t)(b)$ is given by $({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_b)(x)$.

Proof 4.2.

The first part of the proof is standard and works in general but we have described it in full for the sake of the second statement of the lemma which will be used in the subsequent lemma. The map $\sigma :I\times I\to I\times I$ defined by $\sigma (\eta ,x)=(a+\eta (b-a),x)$ is continuous and its composite with ${\chi }_{\gamma ,\kappa (s,t)}$ is given by

$$({\chi }_{\gamma ,\kappa (s,t)}\circ \sigma )(\eta ,x)=H((\kappa (s,t)(x),\gamma (a+\eta (b-a)))).$$

At $x=0$ the corresponding path is $({\chi }_{\gamma ,\kappa (s,t)}\circ \sigma )(\eta ,0)=H((s,\gamma (a+\eta (b-a))))$ whose initial is

$$({\chi }_{\gamma ,\kappa (s,t)}\circ \sigma )(0,0)=H((s,\gamma (a)))=(H\circ {\gamma }_s)(a),$$

and its terminal is

$$({\chi }_{\gamma ,\kappa (s,t)}\circ \sigma )(1,0)=H((s,\gamma (b)))=(H\circ {\gamma }_s)(b).$$

At $x=1$ the corresponding path is $({\chi }_{\gamma ,\kappa (s,t)}\circ \sigma )(\eta ,1)=H((t,\gamma (a+\eta (b-a))))$ whose initial is

$$({\chi }_{\gamma ,\kappa (s,t)}\circ \sigma )(0,1)=H((t,\gamma (a)))=(H\circ {\gamma }_t)(a),$$

and its terminal is

$$({\chi }_{\gamma ,\kappa (s,t)}\circ \sigma )(1,1)=H((t,\gamma (b)))=(H\circ {\gamma }_t)(b).$$

Further we observe that

$$H((\kappa (s,t)(x),\gamma (a+\eta (b-a))))=H((\kappa (s,t)(x),{\gamma }^{\prime }(\eta ))),$$

where the right hand side gives the homotopy from $H\circ {\gamma }_s^{\prime }$ to $H\circ {\gamma }_t^{\prime }$. Now we prove the second part of the lemma. From Remark 3.23 we see that the path which connects $(H\circ {\gamma }_s)(a)=(H\circ {\gamma }_s^{\prime })(0)$ to $(H\circ {\gamma }_t)(a)=(H\circ {\gamma }_t^{\prime })(0)$ is given by

$$p(x)=H((\kappa (s,t)(x),{\gamma }^{\prime }(0)))=H((\kappa (s,t)(x),\gamma (a)))=({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_a)(x),$$

and similarly, the path which connects $(H\circ {\gamma }_s)(b)=(H\circ {\gamma }_s^{\prime })(1)$ to $(H\circ {\gamma }_t)(b)=(H\circ {\gamma }_t^{\prime })(1)$ is given by

$$q(x)=H((\kappa (s,t)(x),{\gamma }^{\prime }(1)))=H(\kappa (s,t)(x),\gamma (b))=({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_b)(x),$$

which concludes the proof.

Lemma 4.3.

The map ${\tilde{\chi }}_{\gamma ,\kappa (s,t)}:I\to {\pi }^l(X\times I)$ given by ${\tilde{\chi }}_{\gamma ,\kappa (s,t)}(\eta )=[{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }]$ is continuous.

Proof 4.4.

Let $N([{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }],𝒰,V,W)$ be any open neighbourhood of $[{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }]$. Here $V$ is an open set belonging to $𝒰$ which contains $({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta })(0)=(H\circ {\gamma }_{\kappa (s,t)(0)})(\eta )$, and similarly, $W$ is an open set belonging to $𝒰$ which contains $({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta })(1)=(H\circ {\gamma }_{\kappa (s,t)(1)})(\eta )$. It follows from the continuity of $H\circ {\gamma }_{\kappa (s,t)(0)}$ that ${(H\circ {\gamma }_{\kappa (s,t)(0)})}^{-1}(V)$ is an open neighbourhood of $\eta$ in $I$, and similarly from the continuity of $H\circ {\gamma }_{\kappa (s,t)(1)}$ we have that ${(H\circ {\gamma }_{\kappa (s,t)(1)})}^{-1}(W)$ is another open neighbourhood of $\eta$ in $I$. Hence $𝒪={(H\circ {\gamma }_{\kappa (s,t)(0)})}^{-1}(V)\cap {(H\circ {\gamma }_{\kappa (s,t)(1)})}^{-1}(W)$ is an open neighbourhood of $\eta$ in $I$. There is an open interval ${ℐ}_{\eta }\subseteq 𝒪$ which contains $\eta$. We prove now that ${\tilde{\chi }}_{\gamma ,\kappa (s,t)}({ℐ}_{\eta })\subseteq N([{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }],𝒰,V,W)$ from which the continuity of ${\tilde{\chi }}_{\gamma ,\kappa (s,t)}$ follows. Let ${\eta }^{\prime }\in {ℐ}_{\eta }$ be arbitrary and assume that . We observe that for every

$$(H\circ {\gamma }_{\kappa (s,t)(0)})({\eta }^{\prime \prime })\in (H\circ {\gamma }_{\kappa (s,t)(0)})({ℐ}_{\eta })\subseteq (H\circ {\gamma }_{\kappa (s,t)(0)})({(H\circ {\gamma }_{\kappa (s,t)(0)})}^{-1}(V))=V,$$

which proves that the image of the restriction of $H\circ {\gamma }_{\kappa (s,t)(0)}$ in $[\eta ,{\eta }^{\prime }]$ lies inside $V$. Similarly, we see that

$$(H\circ {\gamma }_{\kappa (s,t)(1)})({\eta }^{\prime \prime })\in (H\circ {\gamma }_{\kappa (s,t)(1)})({ℐ}_{\eta })\subseteq (H\circ {\gamma }_{\kappa (s,t)(1)})({(H\circ {\gamma }_{\kappa (s,t)(1)})}^{-1}(W))=W,$$

which implies that the image of the restriction of $H\circ {\gamma }_{\kappa (s,t)(1)}$ in $[\eta ,{\eta }^{\prime }]$ lies inside $W$. From Lemma 3.21 the restrictions of the paths $H\circ {\gamma }_{\kappa (s,t)(0)}$ and $H\circ {\gamma }_{\kappa (s,t)(1)}$ in $[\eta ,{\eta }^{\prime }]$ are homotopic, and from Lemma 4.1, the homotopy class of the path connecting $(H\circ {\gamma }_{\kappa (s,t)(0)})(\eta )$ to $(H\circ {\gamma }_{\kappa (s,t)(1)})(\eta )$ is $[{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }]={\tilde{\chi }}_{\gamma ,\kappa (s,t)}(\eta )$. Similarly, the homotopy class of the path connecting $(H\circ {\gamma }_{\kappa (s,t)(0)})({\eta }^{\prime })$ to $(H\circ {\gamma }_{\kappa (s,t)(1)})({\eta }^{\prime })$ is $[{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}]={\tilde{\chi }}_{\gamma ,\kappa (s,t)}({\eta }^{\prime })$. Lemma 3.24 implies that

$${({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta })}^{-1}\ast (H\circ {\gamma }_{\kappa (s,t)(0)}^{\prime })\ast ({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{{\eta }^{\prime }})\sim (H\circ {\gamma }_{\kappa (s,t)(1)}^{\prime }),$$

whence

$$({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{{\eta }^{\prime }})\sim {(H\circ {\gamma }_{\kappa (s,t)(0)}^{\prime })}^{-1}\ast ({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta })\ast (H\circ {\gamma }_{\kappa (s,t)(1)}^{\prime }).$$

But, from the first part of the lemma, $Im(H\circ {\gamma }_{\kappa (s,t)(0)}^{\prime })\subseteq V$ and $Im(H\circ {\gamma }_{\kappa (s,t)(1)}^{\prime })\subseteq W$, therefore ${\tilde{\chi }}_{\gamma ,\kappa (s,t)}({\eta }^{\prime })=[{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}]\in N([{\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }],𝒰,V,W)$.

Lemma 4.5.

For every path $\gamma$, and every $s,t,\eta \in I$, ${\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta }={({\chi }_{\gamma ,\kappa (t,s)}\circ {\mathrm{\ell }}_{\eta })}^{-1}$.

Proof 4.6.

For every $x\in I$ we have that

${({\chi }_{\gamma ,\kappa (t,s)}\circ {\mathrm{\ell }}_{\eta })}^{-1}(x)$ $=({\chi }_{\gamma ,\kappa (t,s)}\circ {\mathrm{\ell }}_{\eta })(1-x)$

$=H((\kappa (t,s)(1-x),\gamma (\eta )))$

$=H((\kappa (s,t)(x),\gamma (\eta )))$

$=({\chi }_{\gamma ,\kappa (s,t)}\circ {\mathrm{\ell }}_{\eta })(x),$

therefore we have the equality.

Lemma 4.7.

Let $Y$ be any connected and locally path connected space and let ${\pi }^l(Y)$ be its fundamental groupoid equipped with Lasso topology. The map $𝛊:{\pi }^l(Y)\to {\pi }^l(Y)$ defined by $[p]\mapsto [p^{-1}]$ is continuous.

Proof 4.8.

The claim is an immediate consequence of the fact that in topological groupoids such as ${\pi }^l(Y)$, the inverse map $𝛊$ is continuous.

Lemma 4.9.

The map $𝛊:{\pi }^l(X\times J)\to {\pi }^l(X\times J)$ induces a covariant functor $\pi (𝛊):\pi ({\pi }^l(X\times J))\to \pi ({\pi }^l(X\times J))$ of groupoids.

Proof 4.10.

This is an immediate consequence of the fact that there is a covariant functor $\pi :\mathrm{𝐓𝐨𝐩}\to \mathrm{𝐆𝐫𝐩𝐝}$ which maps any space $X$ to its fundamental groupoid $\pi X$ and any mapping of spaces $f:X\to Y$ to the morphism of groupoids $\pi (f):\pi X\to \pi Y$ defined by $\pi f([\alpha ])=[f\alpha ]$ (see 6.4.2 of [2]). In our case, for the map $𝛊:{\pi }^l(X\times J)\to {\pi }^l(X\times J)$, we derive that the corresponding morphism of groupoids $\pi (𝛊):\pi ({\pi }^l(X\times J))\to \pi ({\pi }^l(X\times J))$ maps every homotopy class $[p]$ of a path $p$ in ${\pi }^l(X\times J)$ to $[𝛊p]$.

Before we make the next definition we recall two facts from groupoids. First, any nonempty set $X$ gives rise to the discrete groupoid $X$ whose objects are the elements of $X$ and for every $x\in X$ there is a single morphism $1_x$ (see [2], Example 1, p. 217). Second, for every two groupoids $X$ and $Y$, we have their direct product groupoid $X\times Y$ with object set $X\times Y$ and morphisms $(\alpha ,\beta ):(x,y)\to (x^{\prime },y^{\prime })$ wherever $\alpha :x\to x^{\prime }$ is a morphism in $X$ and $\beta :y\to y^{\prime }$ is a morphism in $Y$ (see [2], p. 222).

Definition 4.11.

Let $X$ be the discrete groupoid arising from a nonempty set $X$ and $X\times {\pi }^l(X\times J)$ the direct product groupoid. For every fuzzy subset $F$ in $X$ we define $[F]:X\times {\pi }^l(X\times J)\to \pi ({\pi }^l(X\times J))$ on objects by $[F](y,(z,\beta ))=[p_{F,(y,(z,\beta ))}]$ where $p_{F,(y,(z,\beta ))}$ is the path in $X\times J$ given by $p_{F,(y,(z,\beta ))}(u)=H((\kappa (F(y),1-F(y))(u),(z,\beta )))$, and on morphisms by $[F](y,[\gamma ])=[{\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}]$.

Lemma 4.12.

For every fuzzy subset $G$ in $X$ and every $(y,[c_{(z,\beta )}])\in X\times {\pi }^l(X\times J)$, the path ${\tilde{\chi }}_{c_{(z,\beta )},\kappa (G(y),1-G(y))}$ is the constant path $c_{[p_{G,(y,(z,\beta ))}]}$ at $[p_{G,(y,(z,\beta ))}]$.

Proof 4.13.

For every $\eta \in I$,

$$({\tilde{\chi }}_{c_{(z,\beta )},\kappa (G(y),1-G(y))})(\eta )=[{\chi }_{c_{(z,\beta )},\kappa (G(y),1-G(y))}\circ {\mathrm{\ell }}_{\eta }].$$ (10)

To prove the claim we show that paths ${\chi }_{c_{(z,\beta )},\kappa (G(y),1-G(y))}\circ {\mathrm{\ell }}_{\eta }$ in $X\times J$ with $\eta$ varying in $I$ coincide. Indeed, for every $u\in I$,

$({\chi }_{c_{(z,\beta )},\kappa (G(y),1-G(y))}\circ {\mathrm{\ell }}_{\eta })(u)$ $=H((\kappa (G(y),1-G(y))(u),c_{(z,\beta )}(\eta )))$

$=H((\kappa (G(y),1-G(y))(u),(z,\beta ))),$ (11)

where the right hand side is independent of $\eta$. The second part of the lemma follows from (10) and (11).

Proposition 4.14.

For every fuzzy subset $F$ in $X$, $[F]$ is a covariant functor of groupoids.

Proof 4.15.

First we prove the correctness of $[F]$. Let $\gamma \sim \delta$ and we want to show that for every $y\in X$, ${\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}\sim {\tilde{\chi }}_{\delta ,\kappa (F(y),1-F(y))}$. If $\phi :I\times I\to X\times J$ is a homotopy between $\gamma$ and $\delta$, then for every $t\in I$, the restriction $\phi (\bullet ,t)$ of $\phi$ on $I\times \{t\}$ is a path in $X\times J$ from $\gamma (0)=\delta (0)$ to $\gamma (1)=\delta (1)$. We let $\tilde{\phi }:I\times I\to {\pi }^l(X\times J)$ be the map given by

$$\tilde{\phi }(\eta ,t)={\tilde{\chi }}_{\phi (\bullet ,t),F(y),1-F(y)}(\eta )=[{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }].$$

This map is a homotopy between ${\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}$ and ${\tilde{\chi }}_{\delta ,\kappa (F(y),1-F(y))}$. Indeed,

$$\tilde{\phi }(\eta ,0)=[{\chi }_{\phi (\bullet ,0),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }]=[{\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }]=({\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})(\eta ),$$

and

$$\tilde{\phi }(\eta ,1)=[{\chi }_{\phi (\bullet ,1),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }]=[{\chi }_{\delta ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }]=({\tilde{\chi }}_{\delta ,\kappa (F(y),1-F(y))})(\eta ).$$

Also for every $t\in I$,

$$\tilde{\phi }(0,t)=[{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0]=[{\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0]=({\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})(0),$$

where the second equality holds true since for every $u\in I$,

$({\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0)(u)$ $=H((\kappa (F(y),1-F(y))(u),\phi (0,t)))$

$=H((\kappa (F(y),1-F(y))(u),\gamma (0)))$

$=({\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0)(u).$

In a similar way one can check that

$$\tilde{\phi }(1,t)=({\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})(1).$$

It remains now to prove the continuity of $\tilde{\phi }$. Let $(\eta ,t)\in I\times I$ be arbitrary and $N([{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }],𝒰,V,W)$ be any open neighbourhood of $[{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }]$ in ${\pi }^l(X\times J)$. It follows that $V$ is an open neighbourhood of

$$({\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })(0)=H((F(y),\phi (\eta ,t))),$$

and that $W$ is an open neighbourhood of

$$({\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })(1)=H((1-F(y),\phi (\eta ,t))).$$

The continuity of $H$ implies that there is an open $A$ in $I$ and an open $B$ in $X\times J$ such that $F(y)\in A$, $\phi (\eta ,t)\in B$ and that $A\times B\subseteq H^{-1}(V)$. Similarly, there is an open $A^{\prime }$ in $I$ and an open $B^{\prime }$ in $X\times J$ such that $1-F(y)\in A^{\prime }$, $\phi (\eta ,t)\in B^{\prime }$ and that $A^{\prime }\times B^{\prime }\subseteq H^{-1}(W)$. Consequently $B\cap B^{\prime }$ is an open neighbourhood of $\phi (\eta ,t)$ in $X\times J$. The continuity of $\phi$ implies that ${\phi }^{-1}(B\cap B^{\prime })$ is an open neighbourhood of $(\eta ,t)$ in $I\times I$. We can chose some $r>0$ such that the open ball $B((\eta ,t),r)$ with center $(\eta ,t)$ and radius $r$ is included in ${\phi }^{-1}(B\cap B^{\prime })$. We prove that $\tilde{\phi }(B((\eta ,t),r))\subseteq N([{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }],𝒰,V,W)$ which implies the continuity of $\tilde{\phi }$. Let $({\eta }^{\prime },t^{\prime })\in B((\eta ,t),r)$ and want to prove that

$$\tilde{\phi }({\eta }^{\prime },t^{\prime })=[{\chi }_{\phi (\bullet ,t^{\prime }),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}]\in N([{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }],𝒰,V,W).$$ (12)

The homotopy $\phi$ induces a path $\phi ({\eta }^{\prime },\bullet ):I\to X\times J$ by the rule $t^{\prime \prime }\mapsto \phi ({\eta }^{\prime },t^{\prime \prime })$ which in turn induces two paths, $H\circ \phi {({\eta }^{\prime },\bullet )}_{F(y)}$ and $H\circ \phi {({\eta }^{\prime },\bullet )}_{1-F(y)}$ which are freely homotopic. Under the assumption that , Lemma 4.1 implies that the restrictions $\alpha$ and ${\alpha }^{\prime }$ of respectively $H\circ \phi {({\eta }^{\prime },\bullet )}_{F(y)}$ and $H\circ \phi {({\eta }^{\prime },\bullet )}_{1-F(y)}$ on $[t^{\prime },t]$ are also freely homotopic, and the path which connects $H((F(y),\phi ({\eta }^{\prime },t^{\prime })))$ to $H((1-F(y),\phi ({\eta }^{\prime },t^{\prime })))$ is ${\chi }_{\phi ({\eta }^{\prime },\bullet ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{t^{\prime }}$, and the path which connects $H((F(y),\phi ({\eta }^{\prime },t)))$ to $H((1-F(y),\phi ({\eta }^{\prime },t)))$ is ${\chi }_{\phi ({\eta }^{\prime },\bullet ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_t$. It is easy to see that

$${\chi }_{\phi ({\eta }^{\prime },\bullet ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{t^{\prime }}={\chi }_{\phi (\bullet ,t^{\prime }),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }},$$

and

$${\chi }_{\phi ({\eta }^{\prime },\bullet ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_t={\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}.$$

It follows from Lemma 3.24 that

$${\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}\sim {\alpha }^{-1}\ast ({\chi }_{\phi (\bullet ,t^{\prime }),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }})\ast {\alpha }^{\prime }.$$ (13)

We prove that $\alpha$ is included in $V$ and that ${\alpha }^{\prime }$ is included in $W$. We present here the proof for the first claim since the proof for the second is dual. Any point of $\alpha$ is of the form $H((F(y),\phi ({\eta }^{\prime },t^{\prime \prime })))$ with $t^{\prime \prime }\in [t^{\prime },t]$ where $({\eta }^{\prime },t^{\prime \prime })\in B((\eta ,t),r)$ since

This implies that

$$\phi ({\eta }^{\prime },t^{\prime \prime })\in \phi (B((\eta ,t),r))\subseteq \phi ({\phi }^{-1}(B\cap B^{\prime }))\subseteq \phi ({\phi }^{-1}(B))=B.$$

But $F(y)\in A$, therefore $(F(y),\phi ({\eta }^{\prime },t^{\prime \prime }))\in A\times B\subseteq H^{-1}(V)$, and then $H((F(y),\phi ({\eta }^{\prime },t^{\prime \prime })))\in H(H^{-1}(V))=V$. Now the facts that ${\alpha }^{-1}\subseteq V$ and that ${\alpha }^{\prime }\subseteq W$ together with (13) imply that

$$[{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}]\in N([{\chi }_{\phi (\bullet ,t^{\prime }),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}],𝒰,V,W).$$ (14)

Further, under the assumption that , Lemma 4.1 implies that the restrictions $\beta$ and ${\beta }^{\prime }$ of respectively $H\circ \phi {(\bullet ,t)}_{F(y)}$ and $H\circ \phi {(\bullet ,t)}_{1-F(y)}$ on $[{\eta }^{\prime },\eta ]$ are homotopic with each other, and the path which connects $H((F(y),\phi ({\eta }^{\prime },t)))$ to $H((1-F(y),\phi ({\eta }^{\prime },t)))$ is ${\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}$, and the path which connects $H(F(y),\phi (\eta ,t))$ to $H(1-F(y),\phi (\eta ,t))$ is ${\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }$. Since

$${({\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }})}^{-1}\ast \beta \ast ({\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })\sim {\beta }^{\prime },$$

it follows that

$$\beta \ast ({\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })\ast {\beta }^{\prime -1}\sim {\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}.$$ (15)

Also we have that $\beta$ is included in $V$ and ${\beta }^{\prime }$ is included in $W$. We present here the proof for ${\beta }^{\prime }$ since the proof for $\beta$ is dual and similar to the proof that $\alpha$ is included in $V$. Any point on ${\beta }^{\prime }$ has the form $H((1-F(y),\phi ({\eta }^{\prime \prime },t)))$ with ${\eta }^{\prime \prime }\in [{\eta }^{\prime },\eta ]$ and $({\eta }^{\prime \prime },t)\in B((\eta ,t),r)$ since

This implies that

$$\phi ({\eta }^{\prime \prime },t)\in \phi (B((\eta ,t),r))\subseteq \phi ({\phi }^{-1}(B\cap B^{\prime }))\subseteq \phi ({\phi }^{-1}(B^{\prime }))=B^{\prime }.$$

This, together with $1-F(y)\in A^{\prime }$, imply that $(1-F(y),\phi ({\eta }^{\prime \prime },t))\in A^{\prime }\times B^{\prime }\subseteq H^{-1}(W)$, consequently $H((1-F(y),\phi ({\eta }^{\prime \prime },t)))\in H(H^{-1}(W))=W$. Now (15) and the facts that $\beta \subseteq V$ and ${\beta }^{\prime }\subseteq W$ imply that

$$[{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{{\eta }^{\prime }}]\in N([{\chi }_{\phi (\bullet ,t),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }],𝒰,V,W).$$ (16)

Now (14), (16) and Lemma 2.3 of [6] imply (12) as desired.

To check the functoriality of $[F]$ we first see that $[F]$ sends identities to identities. Indeed, for every object $(y,(z,\beta ))$,

$[F](e_{(y,(z,\beta ))})$ $=[F](y,[c_{(z,\beta )}])$

$=[{\tilde{\chi }}_{c_{(z,\beta )},\kappa (F(y),1-F(y))}]$

$=[c_{[p_{F,(y,(z,\beta ))}]}]$ $(\text{by Lemma }\text{<a href="\#S4.Thmtheorem12" title="Lemma 4.12. ‣ 4. Fuzzy complements in topological terms ‣ A groupoid approach to the study of fuzzy topological spaces" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">4.12</span></a>})$

$=[c_{[F](y,(z,\beta ))}]$ $(\text{by the definition})$

$=e_{[F](y,(z,\beta ))}$

Let now $[\gamma ],[\delta ]\in {\pi }^l(X\times J)$ such that $[\gamma ]\ast [\delta ]$ exists, then for every $y\in X$, $(y,[\gamma ])\ast (y,[\delta ])=(y,[\gamma ]\ast [\delta ])=(y,[\gamma \ast \delta ])$. Let us check that $[F](y,[\gamma ])=[{\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}]$ is composable with $[F](y,[\delta ])=[{\tilde{\chi }}_{\delta ,\kappa (F(y),1-F(y))}]$. This follows if we prove that $({\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})(1)=({\tilde{\chi }}_{\delta ,\kappa (F(y),1-F(y))})(0)$, or equivalently that ${\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_1={\chi }_{\delta ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0$. This is indeed so since for every $x\in I$ we have that

$({\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_1)(x)$ $=H((\kappa (F(y),1-F(y))(x),\gamma (1)))$

$=H((\kappa (F(y),1-F(y))(x),\delta (0)))$ $(\gamma \ast \delta \text{ exists})$

$=({\chi }_{\delta ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0)(x).$

Now we have to prove that

$$[F](y,[\gamma ])\ast [F](y,[\delta ])=[F](y,[\gamma \ast \delta ]),$$

or equivalently that

$$[{\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}]\ast [{\tilde{\chi }}_{\delta ,\kappa (F(y),1-F(y))}]=[{\tilde{\chi }}_{\gamma \ast \delta ,\kappa (F(y),1-F(y))}].$$

We prove this by proving that

$$({\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})\ast ({\tilde{\chi }}_{\delta ,\kappa (F(y),1-F(y))})={\tilde{\chi }}_{(\gamma \ast \delta ),\kappa (F(y),1-F(y))}.$$

The path in the left hand side of the above is given by

Observe that for every $x\in I$, if $0\le \eta \le 1/2$, then

$$({\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{2\eta })(x)=H((\kappa (F(y),1-F(y))(x),\gamma (2\eta ))),$$

and when $1/2\le \eta \le 1$, then

$$({\chi }_{\delta ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{2\eta -1})(x)=H((\kappa (F(y),1-F(y))(x),\delta (2\eta -1))).$$

On the other hand, since

$$({\tilde{\chi }}_{(\gamma \ast \delta ),\kappa (F(y),1-F(y))})(\eta )=[{\chi }_{(\gamma \ast \delta ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }],$$

we have that for every $x\in I$,

$$({\chi }_{(\gamma \ast \delta ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })(x)=H((\kappa (F(y),1-F(y))(x),(\gamma \ast \delta )(\eta ))).$$

It follows that for $0\le \eta \le 1/2$,

$$({\chi }_{(\gamma \ast \delta ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })(x)=H((\kappa (F(y),1-F(y))(x),\gamma (2\eta ))),$$

and for $1/2\le \eta \le 1$,

$$({\chi }_{(\gamma \ast \delta ),\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })(x)=H((\kappa (F(y),1-F(y))(x),\delta (2\eta -1))).$$

By comparing we get the desired equality.

Theorem 4.16.

• For every two fuzzy subsets $F$ and $G$ in $X$, $\pi (𝜾)\circ [F]=[G]$ if and only if $G=1-F$.

• For every fuzzy subset $F$ in $X$, for every $y\in X$ and every $(z,\beta )\in X\times J$ we have in particular that $[1-F]((y,(z,\beta )))={([F]((y,(z,\beta ))))}^{-1}$.

Proof 4.17.

(i) Assume that $G=1-F$ and let $(y,[\gamma ])\in X\times {\pi }^l(X\times J)$ be arbitrary. Then,

$$(\pi (𝜾)\circ [F])((y,[\gamma ]))=\pi (𝜾)([{\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}])=[𝜾\circ {\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}],$$

and $[1-F]((y,[\gamma ]))=[{\tilde{\chi }}_{\gamma ,\kappa (1-F(y),F(y))}]$. The claim follows if prove that for every $\eta \in I$,

$$(𝜾\circ {\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})(\eta )=({\tilde{\chi }}_{\gamma ,\kappa (1-F(y),F(y))})(\eta ).$$

But on the one side

$$(𝜾\circ {\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})(\eta )=𝜾([{\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }])=[{({\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })}^{-1}],$$

and on the other side

$$({\tilde{\chi }}_{\gamma ,\kappa (1-F(y),F(y))})(\eta )=[{\chi }_{\gamma ,\kappa (1-F(y),F(y))}\circ {\mathrm{\ell }}_{\eta }].$$

From Lemma 4.5 we have that ${({\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta })}^{-1}={\chi }_{\gamma ,\kappa (1-F(y),F(y))}\circ {\mathrm{\ell }}_{\eta }$, therefore the equality follows.

Conversely, assume that $\pi (𝛊)\circ [F]=[G]$, therefore for every $(y,[\gamma ])\in X\times {\pi }^l(X\times J)$ we have that

$$[{\tilde{\chi }}_{\gamma ,\kappa (G(y),1-G(y))}]=[G]((y,[\gamma ]))=(\pi (𝜾)\circ [F])((y,[\gamma ]))=[𝜾\circ {\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}].$$

This implies that for $\eta =0$ in particular we have,

$$[{\chi }_{\gamma ,\kappa (G(y),1-G(y))}\circ {\mathrm{\ell }}_0]=({\tilde{\chi }}_{\gamma ,\kappa (G(y),1-G(y))})(0)=(𝜾\circ {\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))})(0)=[{({\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0)}^{-1}].$$

Taking $x=0$ we obtain

$H((G(y),\gamma (0)))$ $=H((\kappa (G(y),1-G(y))(0),\gamma (0)))$

$=({\chi }_{\gamma ,\kappa (G(y),1-G(y))}\circ {\mathrm{\ell }}_0)(0)$

$={({\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_0)}^{-1}(0)$

$=({\chi }_{\gamma ,\kappa (1-F(y),F(y))}\circ {\mathrm{\ell }}_0)(0)$ $(\text{from Lemma }\text{<a href="\#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4. Fuzzy complements in topological terms ‣ A groupoid approach to the study of fuzzy topological spaces" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">4.5</span></a>})$

$=H((1-F(y),\gamma (0))).$

Since this holds true for every path $\gamma$, we can chose it to satisfy $\gamma (0)=(z,\beta )$ with $\beta \ne 0$, hence we have $(1-G(y))\beta =F(y)\beta$. This implies that for every $y\in X$, $G(y)=1-F(y)$ therefore $G=1-F$.
(ii) Combining part (i) of the theorem together with the definitions of $[F]$ and $\pi (𝛊)$ we have the following

$[1-F]((y,(z,\beta )))$ $=\pi (𝜾)([F]((y,(z,\beta ))))$ $(\text{by part (i) of the theorem})$

$=\pi (𝜾)([p_{F,(y,(z,\beta ))}])$ $(\text{by definition }\text{<a href="\#S4.Thmtheorem11" title="Definition 4.11. ‣ 4. Fuzzy complements in topological terms ‣ A groupoid approach to the study of fuzzy topological spaces" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">4.11</span></a>})$

$=[p_{F,(y,(z,\beta ))}^{-1}]$ $(\text{definition of }\pi (𝜾))$

$={([F]((y,(z,\beta ))))}^{-1}$ $(\text{by definition }\text{<a href="\#S4.Thmtheorem11" title="Definition 4.11. ‣ 4. Fuzzy complements in topological terms ‣ A groupoid approach to the study of fuzzy topological spaces" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">4.11</span></a>}.)$

This completes the proof.

We will conclude this section by giving an example whose aim is to make the statement of the above theorem digestible.

Example 4.18.

Let $𝒯=\{\theta ,1,{\displaystyle \frac{1}{3}}\}$ be the fuzzy topology on any nonempty set $X$ considered in Counterexample 2.1 and let $(X\times J,\iota (𝒯))$ be the corresponding topological space. For every segment $[a,b]\subset J$ and every $z\in X$ we know from Remark 3.23 that there is a path $\gamma$ in $X\times J$ from $(z,a)$ to $(z,b)$. For every fuzzy subset $F$ in $X$ and every $y\in X$, we know that $[F]((y,[\gamma ]))$ is represented in $\pi ({\pi }^l(X\times J))$ by the path ${\tilde{\chi }}_{\gamma ,\kappa (F(y),1-F(y))}$ whose points in ${\pi }^l(X\times J)$ are the homotopy classes $[{\chi }_{\gamma ,\kappa (F(y),1-F(y))}\circ {\mathrm{\ell }}_{\eta }]$ with $\eta$ varying in $I$. Taking $F$ to be the constant map ${\displaystyle \frac{1}{3}}$ we see that the corresponding point of $\eta$ in this case is the class $[{\chi }_{\gamma ,\kappa (1/3(y),2/3(y))}\circ {\mathrm{\ell }}_{\eta }]$. We will describe the path ${\chi }_{\gamma ,\kappa (1/3(y),2/3(y))}\circ {\mathrm{\ell }}_{\eta }$ and compare it with ${\chi }_{\gamma ,\kappa (2/3(y),1/3(y))}\circ {\mathrm{\ell }}_{\eta }$ whose class $[{\chi }_{\gamma ,\kappa (2/3,1/3)}\circ {\mathrm{\ell }}_{\eta }]$ is the point corresponding to $\eta$ of ${\tilde{\chi }}_{\gamma ,\kappa (2/3(y),1/3(y))}$ which represents $[2/3]((y,[\gamma ]))$ in $\pi ({\pi }^l(X\times J))$. We observe that

$({\chi }_{\gamma ,\kappa (1/3(y),2/3(y))}\circ {\mathrm{\ell }}_{\eta })(t)$ $=H((\kappa (1/3,2/3)(t),\gamma (\eta )))$

$=H(((t+1)/3,(z,(b-a)\eta +a)))$

$=(z,(2-t)((b-a)\eta +a)/3).$

In a similar fashion we find that

$$({\chi }_{\gamma ,\kappa (2/3(y),1/3(y))}\circ {\mathrm{\ell }}_{\eta })(t)=(z,(1+t)((b-a)\eta +a)/3).$$

But paths ${\chi }_{\gamma ,\kappa (1/3(y),2/3(y))}\circ {\mathrm{\ell }}_{\eta }$ and ${\chi }_{\gamma ,\kappa (2/3(y),1/3(y))}\circ {\mathrm{\ell }}_{\eta }$ are inverses of each other since for each $t\in I$,

$$({\chi }_{\gamma ,\kappa (1/3(y),2/3(y))}\circ {\mathrm{\ell }}_{\eta })(1-t)=({\chi }_{\gamma ,\kappa (2/3(y),1/3(y))}\circ {\mathrm{\ell }}_{\eta })(t).$$

This shows that the point of ${\tilde{\chi }}_{\gamma ,\kappa ((1/3)(y),(1-1/3)(y))}$ corresponding to $\eta$ is the inverse of the point of
${\tilde{\chi }}_{\gamma ,\kappa ((1-1/3)(y),(1/3)(y))}$ corresponding to $\eta$, which was expectable since $\pi (𝛊)\circ [F]=[1-F]$ and $\pi (𝛊)$ maps point to their inverses. This new interpretation of the duality between the complementary maps $1/3$ and $2/3$ is in contrast with Example 2.1 where for the coarser topology ${\mathrm{\Psi }}^{\ast }(𝒯)$ we were not able to find any set theoretic relationship between ${\mathrm{\Psi }}^{\ast }(2/3)$ and ${\mathrm{\Psi }}^{\ast }(1/3)$ apart from the observation that ${\mathrm{\Psi }}^{\ast }(2/3)\ne (X\times J)\setminus {\mathrm{\Psi }}^{\ast }(1/3)$.