Abstract.

We show that, given a semitopological group $G$, the semitopological groups $T_i(G^{\bullet })$ and ${(T_i(G))}^{\bullet }$, for each $i\in \{0,1,2\}$, and ${(G_{sr})}^{\bullet }$ and ${(G^{\bullet })}_{sr}$, are topologically isomorphic. Moreover, we prove that, given a paratopological group $G$, the paratopological groups $Reg(G^{\bullet })$ and ${(Reg(G))}^{\bullet }$ are topologically isomorphic.

1. Introduction

Given a topological (paratopological, semitopological) group $G$, we can construct other topological (paratopological, semitopological) groups associated with $G$, possibly with some improvements in their topological properties. In [2], S. Hartman and J. Mycielski presented a functorial construction that permits to embed a Hausdorff topological group into a path connected, locally path connected Hausdorff topological group. A brief outline of their construction is given below.

Let $G$ be a topological group with identity $e$. Consider the set $G^{\bullet }$ of all functions $f$ on $J=[0,1)$ with values in $G$ such that, for some sequence , the function $f$ is constant on $[a_k,a_{k+1})$ for each $k=0,1,\mathrm{\dots },n-1$.

Define a binary operation $\ast$ on $G^{\bullet }$ by $(f\ast g)(x)=f(x)g(x)$, for all $f,g\in G^{\bullet }$ and $x\in J$. Each element $f\in G^{\bullet }$ has a unique inverse $f^{-1}\in G^{\bullet }$ defined by $(f^{-1})(x)={(f(x))}^{-1}$, for each $x\in J$. Then $(G^{\bullet },\ast )$ is a group with identity $e^{\bullet }$, where $e^{\bullet }(x)=e$ for each $x\in J$.

Let $V$ be an open neighborhood of $e$ in $G$ and $ϵ$ a positive real number. Consider the subset

of $G^{\bullet }$, where $\mu$ is the usual Lebesgue measure on $J$. One can verify that the subsets $O(V,ϵ)$ form a base of a Hausdorff topological group topology at the identity of $G^{\bullet }$, thus making $G^{\bullet }$ into a pathwise connected and locally pathwise connected topological group. For details, the reader is referred to [2].

Let $f\in G^{\bullet }$. Then there exist real numbers $a_0,a_1,\mathrm{\dots },a_n$ with such that $f$ is constant on $[a_k,a_{k+1})$ for each $k=0,1,\mathrm{\dots },n-1$. Assuming that $f(a_k)\ne f(a_{k+1})$ for each $k\le n-1$, we put $P(f)=\{a_0,a_1,\mathrm{\dots },a_{n-1}\}$, and $V(f)=\{f(a_0),\mathrm{\dots },f(a_{n-1})\}$. For the identity $e_G^{\bullet }$ of the group $G^{\bullet }$ we put $P(e_G^{\bullet })=\{a_0\}$.

It is easy to verify that in the Hartman–Mycielski construction, if $G$ is a paratopological (semitopological) group, then $G^{\bullet }$ is also a paratopological (semitopological) group. For more information on the Hartman–Mycielski construction, see [1, Section 3.8] and [4]. In what follows, $G^{\bullet }$ refers to the Hartman–Mycielski group associated to $G$ and $𝒩(e)$ to the family of all open neighborhoods of the identity in $G$.

In [8], M. Tkachenko defines what we will call the $T_i$-reflection, $T_i(G)$, of an arbitrary semitopological group $G$ for $i\in \{0,1,2,3,3.5\}$, as well as the regular reflection, $Reg(G)$, of $G$. All these objects are new semitopological groups, and their names indicate that $T_i(G)$ satisfies the $T_i$ separation axiom for $i\in \{0,1,2,3,3.5\}$, whereas $Reg(G)$ is regular, i.e., satisfies the $T_1$ and $T_3$ separation axioms.

In this paper we show that, in the category of semitopological groups, the functors ^∙ and $T_i$ for $i\in \{0,1,2\}$ commute, that is, the semitopological groups $T_i(G^{\bullet })$ and ${(T_i(G))}^{\bullet }$ for $i\in \{0,1,2\}$ are topologically isomorphic. Likewise, we prove that ${(G_{sr})}^{\bullet }$ and ${(G^{\bullet })}_{sr}$ are topologically isomorphic. Finally, we show that in the category of paratopological groups, the functors $Reg$ and ^∙ commute, that is, the paratopological groups $Reg(G^{\bullet })$ and ${(Reg(G))}^{\bullet }$ are topologically isomorphic.

First, we present some simple but very important properties of the Hartman–Mycielski group $G^{\bullet }$ associated to a semitopological group $G$. We omit their proofs, since they are identical to those found in [1, Theorem 3.8.3], [1, Proposition 3.8.6], and [1, Proposition 3.8.7], respectively.

For each $g\in G$, denote by $g^{\bullet }$ the element of $G^{\bullet }$ defined by $g^{\bullet }(r)=g$, for all $r\in J$. Let $i_G:G\to G^{\bullet }$ be the function defined by $i_G(g)=g^{\bullet }$ for each $g\in G$.

Theorem 1.1.

Let $G$ be a semitopological group. Then $i_G:G\to G^{\bullet }$ is a topological monomorphism. If $G$ is Hausdorff, then the image $i_G(G)$ is a closed subgroup of $G^{\bullet }$.

Proposition 1.2.

Let $\phi :G\to H$ be a continuous homomorphism of semitopological groups. Then $\phi$ admits a natural extension to a continuous homomorphism ${\phi }^{\bullet }:G^{\bullet }\to H^{\bullet }$. If $\phi$ is open and onto, then so is ${\phi }^{\bullet }$.

The correspondences $G\mapsto G^{\bullet }$ and $\phi \mapsto {\phi }^{\bullet }$ are functorial since the equality ${(\psi \circ \phi )}^{\bullet }={\psi }^{\bullet }\circ {\phi }^{\bullet }$ holds for any continuous homomorphisms $\phi :G\to H$ and $\psi :H\to K$. This defines the covariant functor ^∙ in the category of semitopological groups and continuous homomorphisms. It turns out that this functor preserves subgroups and quotient groups:

Proposition 1.3.

Let $G$ be a semitopological group and $H$ a subgroup of $G$.

1. If $\phi$ is the identity embedding of $H$ to $G$, then the natural homomorphic extension ${\phi }^{\bullet }:H^{\bullet }\to G^{\bullet }$ is a topological monomorphism.

2. If $H$ is closed or invariant in $G$, then $H^{\bullet }$ is also closed or invariant in $G^{\bullet }$.

3. If $H$ is a closed invariant subgroup of $G$, then the groups ${(G/H)}^{\bullet }$ and $G^{\bullet }/H^{\bullet }$ are naturally topologically isomorphic.

Following [8], we say that a class $𝒞$ of spaces is a $PS$-class if it contains arbitrary products of its elements, is hereditary with respect to taking subspaces, and contains a one-point space. It is clear that the class ${𝒯}_i$ of $T_i$-spaces is a $PS$-class for each $i\in \{0,1,2,3,3.5\}$, and so are the classes of regular spaces (i.e., $T_1$ and $T_3$) and Tychonoff spaces (i.e., $T_1$ and $T_{3.5}$).

Given a semitopological group $G$ and a number $i\in \{0,1,2,3,3.5\}$, the $T_i$-reflection of $G$, denoted by $T_i(G)$, is a pair $(H,{\phi }_{G,i})$, where $H$ is a semitopological group satisfying the $T_i$ separation axiom and ${\phi }_{G,i}$ is a continuous homomorphism from $G$ onto $H$ with the following property: For every continuous mapping $f:G\to X$, where $X$ is a space satisfying the $T_i$ separation axiom, there exists a continuous mapping $h:H\to X$ such that $f=h\circ {\phi }_{G,i}$.

In [8], M. Tkachenko provides an internal characterization of the kernel of the canonical homomorphism ${\phi }_{G,i}$ for $i\in \{0,1\}$. I.  Sánchez, in [6] describes the kernel of the canonical homomorphism ${\phi }_{G,2}$, for any semitopological group $G$.

The following example shows that the functors $T_1$ and $T_3$ do not commute in the category of paratopological groups.

Example 1.4 ([8, Example 3.10]).

Let $\mathbb{Z}$ be the additive group of integers. For every integer $n\ge 0$, let $U_n=\{0\}\cup \{m\in \mathbb{Z}:m>n\}.$ It is easy to verify that the family $\{U_n:n\in \mathbb{N}\}$ forms a neighborhood base at zero for a paratopological group topology ${\tau }_1$ on $\mathbb{Z}$, and that the paratopological group $G=(\mathbb{Z},{\tau }_1)$ satisfies the $T_1$ separation axiom. It is also clear that any two nonempty open subsets of $G$ have a nonempty intersection. Since $G$ is a $T_1$-space, we have $T_1(G)=G$ and hence $T_3(T_1(G))=T_3(G).$ By [8, Lemma 3.9], the canonical homomorphism ${\phi }_{G,3}$ of $G$ onto $T_3(G)$ is a bijection. Therefore, $T_3(G)$ is an infinite group (endowed with the anti-discrete topology, see [7, Theorem 2.6]).

On the other hand, $T_1(T_3(G))$ is the trivial group, for [8, Theorem 3.4]. Consequently, the groups $T_1(T_3(G))$ and $T_3(T_1(G))$ are quite different.

2. $T_i$-reflection and the Hartman–Mycielski construction in semitopological groups for $i\in \{0,1,2\}$

In this section we deal with the $T_i$-reflection of $G^{\bullet }$ for $i\in \{0,1,2\}$, where $G$ is a semitopological group. We will prove that $T_i(G^{\bullet })$ and ${\left(T_i(G)\right)}^{\bullet }$ are topologically isomorphic.

The following Lemmas 2.1 and 2.3 are almost immediate from the definition of subsets $O(U,ϵ)$ of $G^{\bullet }$.

Lemma 2.1.

Let $G$ be a semitopological group, $U\in 𝒩(e)$ and $ϵ>0$. Then ${\left[O(U,ϵ)\right]}^{-1}=O(U^{-1},ϵ)$.

Proof 2.2.

Let $f\in {\left[O(U,ϵ)\right]}^{-1}$. If $g=f^{-1}$, then .

Lemma 2.3.

Let $G$ be a semitopological group, and $U$ and $V$ be nonempty subsets of $G$. Then $O(U,ϵ)O(V,\delta )\subset O(UV,ϵ+\delta )$ for all $ϵ,\delta >0$.

Proof 2.4.

If $h\in O(U,ϵ)O(V,\delta )$, then $h=fg$ for some $f\in O(U,ϵ)$ and $g\in O(V,\delta )$. To see that $h\in O(UV,ϵ+\delta )$ note that

$\{r\in J:h(r)\notin UV\}$ $=\{r\in J:f(r)g(r)\notin UV\}$

$\subset \{r\in J:f(r)\notin U\}\cup \{r\in J:g(r)\notin V\}.$

This proves the lemma.

In Lemmas 2.5 and 2.7, we present some properties of the closures of basic open sets in $G^{\bullet }$.

Lemma 2.5.

Let $G$ be a semitopological group, $U\in 𝒩(e)$, and $ϵ>0$. Then the equality $O(\overline{U},ϵ)={\bigcap }_{V\in 𝒩(e)}O(U{V}^{-1},ϵ)$ holds.

Proof 2.6.

We have that $\overline{U}={\bigcap }_{V\in 𝒩(e)}U{V}^{-1}$. Hence, $O(\overline{U},ϵ)\subset O(U{V}^{-1},ϵ)$ for each $V\in 𝒩(e)$, whence it follows that $O(\overline{U},ϵ)\subset {\bigcap }_{V\in 𝒩(e)}O(U{V}^{-1},ϵ)$.

If $f\in {\bigcap }_{V\in 𝒩(e)}O(U{V}^{-1},ϵ)$, then $f\in O(U{V}^{-1},ϵ)$ for each $V\in 𝒩(e)$. Let us verify that the Lebesgue measure of the set $\{r\in J:f(r)\notin \overline{U}\}$ is less than $ϵ$. Since $f\in G^{\bullet }$, there exist real numbers $a_1,a_2,\mathrm{\dots },a_n$ with , and elements $x_0,x_1,\mathrm{\dots },x_n\in G$, such that $f(r)=x_i$ for each $r\in [a_i,a_{i+1})$, where $0\le i\le n$.

If $V(f)\subset \overline{U}$, then $f\in O(\overline{U},ϵ)$. If $V(f)⊈\overline{U}$ and $f(a_i)\notin \overline{U}$ for some $i\le n$, we can find an element $W_i\in 𝒩(e)$ such that $f(a_i)W_i\cap U=\mathrm{\varnothing }$, so $f(a_i)\notin U{W}_i^{-1}$. Let $M=\{i\le n:f(a_i)\notin \overline{U}\}$ and $W={\bigcap }_{i\in M}{W}_i$. Since $f(a_i)\notin U{W}^{-1}$ for each $i\in M$, we have

$$\{r\in J:f(r)\notin \overline{U}\}=\{r\in J:f(r)\notin U{W}^{-1}\}.$$

Therefore, , and we conclude that $f\in O(\overline{U},ϵ)$. This completes the proof.

Lemma 2.7.

Let $G$ be a semitopological group, $U\in 𝒩(e)$ and $ϵ>0$. Then $O(\overline{U},ϵ)\subset \overline{O(U,ϵ)}$.

Proof 2.8.

Let $f\in O(\overline{U},ϵ)$, $W\in 𝒩(e)$ and $\delta >0$. Suppose that $P(f)=\{a_0,\mathrm{\dots },a_n\}$. For each $i\le n$ such that $f(a_i)\in \overline{U}$, we have $f(a_i)W\cap U\ne \mathrm{\varnothing }$, that is, there exists $w_i\in W$ such that $f(a_i)w_i\in U$. Now, defining $g(r)=w_i$ for each $r\in [a_i,a_{i+1})$ if $f(a_i)\in \overline{U}$, and $g(r)=e$ otherwise, we obtain an element $g\in G^{\bullet }$ such that $fg\in fO(W,\delta )\cap O(U,ϵ)$. Therefore, $f\in \overline{O(U,ϵ)}$, that is, $O(\overline{U},ϵ)\subset \overline{O(U,ϵ)}$.

In the following example we show that the inclusion in Lemma 2.7 can be proper.

Example 2.9.

Let $G={\mathbb{Z}}_p$ be the topological group of $p$-adic integers, for any integer $p\ge 2$. Suppose that the equality $\overline{O(U,ϵ)}=O(\overline{U},ϵ)$ holds in $G^{\bullet }$ for all $U\in 𝒩(e)$ and $ϵ>0$. Since $G$ is zero-dimensional, our assumption implies that

$$O(V,ϵ)=O(\overline{V},ϵ)=\overline{O(V,ϵ)},$$

for each open subgroup $V$ of $G$. This is a contradiction, since $G^{\bullet }$ is a connected semitopological group by [1, Theorem 3.8.3]. Therefore, the equality is not true.

It is known (see [7]) that the $T_2$-reflection of an arbitrary topological group $G$ is the quotient topological group $G/N$, where $N$ is the closure of the identity element of $G$, that is, $T_2(G)=G/\overline{\{e\}}$. Furthermore, since in topological groups the separation axiom $T_0$ is equivalent to complete regularity, it follows that for any topological group $G$, the $T_i$-reflections of $G$, for $i=0,1,2,3,3.5$, are equivalent.

Theorem 2.10.

Let $G$ be a topological group. Then $T_2(G^{\bullet })$ and ${(T_2(G))}^{\bullet }$ are topologically isomorphic.

Proof 2.11.

Let $N=\overline{\{e\}}$. Since $T_2(G)=G/N$, we have ${(T_2(G))}^{\bullet }={(G/N)}^{\bullet }$ and, by Proposition 1.3, the groups ${(G/N)}^{\bullet }$ and $G^{\bullet }/N^{\bullet }$ are topologically isomorphic. We also know that $T_2(G^{\bullet })=G^{\bullet }/K$, where $K=\overline{\{e^{\bullet }\}}$. Hence, it suffices to show that $N^{\bullet }=K$, where

$$K=\{f\in G^{\bullet }:e^{\bullet }\in fO(U,ϵ)\text{ for all }U\in 𝒩(e)\text{ and }ϵ>0\}$$

and $N^{\bullet }=\{f\in G^{\bullet }:V(f)\subset N\}$. Let $f\in N^{\bullet }$ and assume that $P(f)=\{a_0,\mathrm{\dots },a_n\}$. Since $V(f)\subset N$ we have $f(a_k)\in N$ for each $k\le n$, so $e\in f(a_k)U$ for all $U\in 𝒩(e)$ and $k\le n$. Therefore, for each $k$, there exists $u_k\in U$ such that $f(a_k)u_k=e$. We can then define $g\in G^{\bullet }$ with $P(f)=P(g)$ and $g(a_k)=u_k$ for $k=0,\mathrm{\dots },n$. Since $g\in O(U,ϵ)$ for all $ϵ>0$ and the equality $fg=e^{\bullet }$ holds, we see that $e^{\bullet }\in fO(U,ϵ)$. Hence $f\in K$, that is, $N^{\bullet }\subset K$.

Conversely, if $f\notin N^{\bullet }$, then there exists $x\in V(f)$ with $x\notin N$. Since $N$ is closed in $G$, there is $V\in 𝒩(e)$ such that $e\notin xV$. Taking the basic open set $W=O(V,\mathrm{\Delta }(f)/2)$, we clearly have $e^{\bullet }\notin fW$, that is, $f\notin K$. Therefore, $K\subset N^{\bullet }$. This completes the proof.

The subsequent Lemmas 2.12, 2.13 and 2.14 follow from [6, Lemma 2.1], [6, Lemma 2.2] and [6, Theorem 2.3], respectively.

Lemma 2.12.

Let $G$ be a semitopological group with identity $e$ and $𝒩(e)$ be the family of all open neighborhoods of the identity in $G$. Then

$$K=\bigcap \{\overline{O(U,ϵ)}:U\in 𝒩(e),ϵ>0\}$$

is a closed invariant subgroup of $G^{\bullet }$.

The subgroup $K$ of $G^{\bullet }$ defined in Lemma 2.12 is used in the following two results.

Lemma 2.13.

Let $G$ be a semitopological group and $U,V\in 𝒩(e)$. Then

$$O(U,ϵ)KO{(V,\delta )}^{-1}=O(U,ϵ)O{(V,\delta )}^{-1},$$

for all positive real numbers $ϵ,\delta$. Moreover, the equality $\overline{O(U,ϵ)}K=\overline{O(U,ϵ)}$ holds for all $U\in 𝒩(e)$ and $ϵ>0$.

Lemma 2.14.

Let $G$ be a semitopological group. Then, $T_2(G^{\bullet })=G^{\bullet }/K$, so $K$ is the kernel of the canonical homomorphism ${\phi }_{G^{\bullet },2}:G^{\bullet }\to T_2(G^{\bullet })$. Moreover, ${\phi }_{G^{\bullet },2}^{-1}\left({\phi }_{G^{\bullet },2}(\overline{O(U,ϵ)})\right)=\overline{O(U,ϵ)},$ for all $U\in 𝒩(e)$ and $ϵ>0$.

In the following theorem we show that the functors $T_2$ and ^∙ commute in the category of semitopological groups.

Theorem 2.15.

Let $G$ be an arbitrary semitopological group. Then $T_2(G^{\bullet })$ and ${(T_2(G))}^{\bullet }$ are topologically isomorphic.

Proof 2.16.

If $G$ is a Hausdorff semitopological group, then $T_2(G)=G$ and $T_2(G^{\bullet })=G^{\bullet }$, hence $T_2(G^{\bullet })={(T_2(G))}^{\bullet }$. Suppose that $G$ is not Hausdorff. By [6, Theorem 2.3], we have $T_2(G)=G/K$, where $K=\bigcap \{\overline{U}:U\in 𝒩(e)\}$. Then $T_2{(G)}^{\bullet }={(G/K)}^{\bullet }$ and by Proposition 1.3, the groups ${(G/K)}^{\bullet }$ and $G^{\bullet }/K^{\bullet }$ are topologically isomorphic. Also, by Lemma 2.12, we have the equality $T_2(G^{\bullet })=G^{\bullet }/H$, where $H=\bigcap \{\overline{O(U,ϵ)}:U\in 𝒩(e),ϵ>0\}$. It is therefore suffices to prove that the groups $K^{\bullet }$ and $H$ coincide. Note that

$$K^{\bullet }=\{f\in G^{\bullet }:V(f)\subset \bigcap _{U\in 𝒩(e)}\overline{U}\}.$$

If $f\in K^{\bullet }$, then $V(f)\subset {\bigcap }_{U\in 𝒩(e)}\overline{U}$, so Lemma 2.7 implies that

$$f\in \bigcap \{O(\overline{U},ϵ):U\in 𝒩(e),ϵ>0\}\subset H.$$

Thus, $K^{\bullet }\subset H$.

Conversely, if $f\notin K^{\bullet }$, then we can find $x\in V(f)$ and $V\in 𝒩(e)$ such that $x\notin \overline{V}$. Then there exists $W\in 𝒩(e)$ such that $x\notin V{W}^{-1}$, which implies $f\notin O(V{W}^{-1},\mathrm{\Delta }(f)/2)$. By Lemma 2.3, we have

$$f\notin O(V,\mathrm{\Delta }(f)/4)O(W^{-1},\mathrm{\Delta }(f)/4)=O(V,\mathrm{\Delta }(f)/4)O{(W,\mathrm{\Delta }(f)/4)}^{-1}.$$

It follows that $f\notin \overline{O(V,\mathrm{\Delta }(f)/4)}$, thus implying that $f\notin H$. Hence, $H\subset K^{\bullet }$. This completes the proof.

Proposition 2.17.

Let $H$ be a subgroup of a semitopological group $G$. Then $H$ is closed in $G$ if and only if $H^{\bullet }=\{f\in G^{\bullet }:V(f)\subset H\}$ is a closed subgroup of $G^{\bullet }$.

Proof 2.18.

By item b) of Proposition 1.3, it suffices to show that if $H^{\bullet }$ is closed in $G^{\bullet }$, then $H$ is closed in $G$. Suppose that $H^{\bullet }$ is closed in $G^{\bullet }$ and take $x\in \overline{H}$. Then, for any $U\in 𝒩(e)$, we have $xU\cap H\ne \mathrm{\varnothing }$, which clearly implies that $x^{\bullet }O(U,ϵ)\cap H^{\bullet }\ne \mathrm{\varnothing }$ for each $ϵ>0$. Hence, $x^{\bullet }\in \overline{H^{\bullet }}$, and so $x^{\bullet }\in H^{\bullet }$. We conclude that $x\in H$ and that $H$ is closed in $G$.

In Lemma 2.19, we describe the $T_1$-reflection of $G^{\bullet }$, for an arbitrary semitopological group $G$.

Lemma 2.19.

Let $G$ be a semitopological group, $K$ be the smallest closed subgroup of $G$, and $F$ the smallest closed subgroup of $G^{\bullet }$. Then $K^{\bullet }=F$, that is, $K^{\bullet }$ is the smallest closed subgroup of $G^{\bullet }$. Moreover, $T_1(G^{\bullet })=G^{\bullet }/K^{\bullet }$.

Proof 2.20.

Let $I=[a,b)$, where . Define a mapping ${\phi }_I:G\to G^{\bullet }$ by

Clearly, ${\phi }_I$ is a topological monomorphism. Let $p:G^{\bullet }\to G^{\bullet }/F$ be the quotient homomorphism. Then $p\circ {\phi }_I:G\to G^{\bullet }/F$ is a continuous homomorphism. Since $G^{\bullet }/F$ is a $T_1$-space, the kernel $\ker (p\circ {\phi }_I)$ of $p\circ {\phi }_I$ is a closed subgroup of $G$. Therefore, $K\subset \ker (p\circ {\phi }_I)$, which implies that ${\phi }_I(K)\subset F$.

For every $f\in K^{\bullet }$, there exist real numbers $a_1,a_2,\mathrm{\dots },a_n$ with such that $f(r)=x_m\in K$ for each $r\in J_m=[a_m,a_{m+1})$, where $0\le m\le n$. Taking $I=J_m$, we have ${\phi }_{J_m}(x_m)\in F$ for all $0\le m\le n$. Therefore,

$$f=\prod _{m=0}^n{\phi }_{J_m}(x_m)\in F.$$

This shows that $K^{\bullet }\subset F$. Since, by Proposition 2.17, $K^{\bullet }$ is a closed subgroup of $G^{\bullet }$ and $F$ is the smallest closed subgroup of $G^{\bullet }$, it follows that $K^{\bullet }=F$. Finally, by [8, Theorem 3.4], we have the equality $T_1(G^{\bullet })=G^{\bullet }/K^{\bullet }$.

From [8, Theorem 3.4], Proposition 1.3 and Lemma 2.19 we obtain the following theorem:

Theorem 2.21.

Let $G$ be a semitopological group. Then $T_1(G^{\bullet })$ and ${(T_1(G))}^{\bullet }$ are topologically isomorphic.

The next two facts follow from [8, Theorem 3.1].

Lemma 2.22.

Let $G$ be a semitopological group with identity $e$. Let also

$$P=\bigcap \{O(U,ϵ):U\in 𝒩(e),ϵ>0\}\subset G^{\bullet }.$$

Then $K=P\cap P^{-1}$ is an invariant subgroup of $G^{\bullet }$. Furthermore, the equality $KO(U,ϵ)=O(U,ϵ)$ holds for all $U\in 𝒩(e)$ and $ϵ>0$.

We use the subgroup $K$ of $G^{\bullet }$ defined in Lemma 2.22 in the subsequent result.

Lemma 2.23.

Let $G$ be a semitopological group. Then $T_0(G^{\bullet })=G^{\bullet }/K$.

Theorem 2.24.

Let $G$ be an arbitrary semitopological group. Then the groups $T_0(G^{\bullet })$ and ${(T_0(G))}^{\bullet }$ are topologically isomorphic.

Proof 2.25.

By [8, Theorem 3.1], the equality $T_0(G)=G/K$ holds, where $K=\bigcap 𝒩(e)\cap {\left(\bigcap 𝒩(e)\right)}^{-1}$. Let $\pi :G\to G/K$ be the quotient homomorphism. Then, by Proposition 1.2, $\pi$ admits an extension to a continuous open homomorphism ${\pi }^{\bullet }:G^{\bullet }\to {(G/K)}^{\bullet }$. Note that

$$\ker ({\pi }^{\bullet })=\{f\in G^{\bullet }:{\pi }^{\bullet }(f)=e^{\prime }\}=\{g\in G^{\bullet }:V(g)\subset K\}=K^{\bullet },$$

where $e^{\prime }$ is the identity of the group ${(G/K)}^{\bullet }$. Then, by [1, Theorem 1.5.13], $G^{\bullet }/K^{\bullet }$ is topologically isomorphic to ${(G/K)}^{\bullet }$. Therefore, the groups $T_0(G^{\bullet })$ and ${(T_0(G))}^{\bullet }$ are topologically isomorphic.

3. Semiregularization and the Hartman–Mycielski construction in semitopological groups

The operation of semiregularization of a topological space was defined by M. Katetov in [3]. A. Ravsky [5] considered this operation in the classes of semitopological and paratopological groups and proved that the semiregularization of a semitopological (paratopological) group is again a semitopological (paratopological) group. Furthermore, the semiregularization of a Hausdorff paratopological group is a regular space.

In this section we deal with semiregularizations of semitopological groups. Let $G$ be a semitopological group with identity $e$. The family

$${\beta }_{ro}=\{{\mathrm{Int}}_G\overline{U}:U\in 𝒩(e)\}$$

serves as a base at the identity $e$ for a topology ${𝒯}_{sr}$ on $G$, which is coarser than the original topology of $G$. The group $G$ with the topology ${𝒯}_{sr}$ is called the semiregularization of $G$ and is usually denoted by $G_{sr}$. We say that $G$ is semiregular if $G=G_{sr}$.

Our aim is to prove that the semitopological groups ${(G^{\bullet })}_{sr}$ and ${\left(G_{sr}\right)}^{\bullet }$ are topologically isomorphic. This requires two auxiliary results.

Lemma 3.1.

Let $G$ be a semitopological group, , and $U\in 𝒩(e)$. Then

$$\mathrm{Int}O(\overline{U},ϵ)=\mathrm{Int}\overline{O(U,ϵ)}.$$

Proof 3.2.

By Lemma 2.7, it suffices to prove that $\mathrm{Int}\overline{O(U,ϵ)}\subset \mathrm{Int}O(\overline{U},ϵ)$. Take an arbitrary element $f\in G^{\bullet }\setminus O(\overline{U},ϵ)$. Then $\mu (\{r\in J:f(r)\notin \overline{U}\})\ge ϵ$. Let $P(f)=\{a_0,\mathrm{\dots },a_n\}$ and $M=\{0\le i\le n:f(a_i)\notin \overline{U}\}$.

Case I. $\mu (\{r\in J:f(r)\notin \overline{U}\})>ϵ$. For each $m\in M$, there exists $W_m\in 𝒩(e)$ such that $f(a_m)W_m\cap U=\mathrm{\varnothing }$. Let $W={\bigcap }_{m\in M}{W}_m$. Then $f(a_m)W\cap U=\mathrm{\varnothing }$ for each $m\in M$. We claim that $fO(W,\delta /2)\cap O(U,ϵ)=\mathrm{\varnothing }$, where $\delta =\mu (\{r\in J:f(r)\notin \overline{U}\})-ϵ>0$. Indeed, suppose that there exists $h\in fO(W,\delta /2)\cap O(U,ϵ)$. Since $h\in fO(W,\delta /2)$, we have . For each $m\in M$,

$$\{r\in [a_m,a_{m+1}):h(r)\in f(a_m)W\}\subset \{r\in J:h(r)\notin U\},$$

whence it follows that

$$\bigcup _{m\in M}\{r\in [a_m,a_{m+1}):h(r)\in f(a_m)W\}\subset \{r\in J:h(r)\notin U\}.$$

By the choice of $\delta$, we have

$$\mu \left(\bigcup _{m\in M}\{r\in [a_m,a_{m+1}):h(r)\in f(a_m)W\}\right)\ge ϵ+\delta /2>ϵ,$$

thus $\mu (\{r\in J:h(r)\notin U\})>ϵ$, which contradicts the assumption. Therefore, $fO(W,\delta /2)\cap O(U,ϵ)=\mathrm{\varnothing }$, that is, $f\notin \overline{O(U,ϵ)}$.

Case II. $\mu (\{r\in J:f(r)\notin \overline{U}\})=ϵ$. Let $V\in 𝒩(e)$, $m\in M$ and $\delta >0$. Choose $a_k\in P(f)$ and $b\in J$ such that $f(a_k)\in \overline{U}$, and . Define an element $g\in G^{\bullet }$ by the rule

Clearly, $g\in O(V,\delta )$. Then $fg\in \{h\in G^{\bullet }:\mu (\{r\in J:h(r)\notin \overline{U}\})>ϵ\}$, so by Case I, we have $fg\notin \overline{O(U,ϵ)}$. Thus $f\notin \mathrm{Int}\overline{O(U,ϵ)}$.

We have shown that $\mathrm{Int}\overline{O(U,ϵ)}\subset O(\overline{U},ϵ)$, which implies the required inclusion $\mathrm{Int}\overline{O(U,ϵ)}\subset \mathrm{Int}O(\overline{U},ϵ)$.

Lemma 3.3.

Let $G$ be a semitopological group, and $U\in 𝒩(e)$. Then

$$O(\mathrm{Int}\overline{U},ϵ)=\mathrm{Int}\overline{O(U,ϵ)}.$$

Proof 3.4.

Since $\mathrm{Int}\overline{U}\subset \overline{U}$ for all $U\in 𝒩(e)$, we have $O(\mathrm{Int}\overline{U},ϵ)\subset O(\overline{U},ϵ)$. Then, by Lemma 3.1, we deduce that $O(\mathrm{Int}\overline{U},ϵ)\subset \mathrm{Int}\overline{O(U,ϵ)}$. Conversely, let $f\in G^{\bullet }\setminus O(\mathrm{Int}\overline{U},ϵ)$, and suppose that $P(f)=\{a_0,a_1,\mathrm{\dots },a_n\}$. Then $\mu (\{r\in J:f(r)\notin \mathrm{Int}\overline{U}\})\ge ϵ$. If $\mu (\{r\in J:f(r)\notin \overline{U}\})\ge ϵ$, then $f\notin O(\overline{U},ϵ)$, hence $f\notin \mathrm{Int}\overline{O(U,ϵ)}$. Otherwise, let

$$M=\{0\le k\le n:f(a_k)\in \overline{U}\setminus \mathrm{Int}\overline{U}\}\ne \mathrm{\varnothing }.$$

It follows that

$$\mu (\{r\in J:f(r)\notin \mathrm{Int}\overline{U}\})=\sum _{k\in M}(a_{k+1}-a_k)+\mu (\{r\in J:f(r)\notin \overline{U}\})\ge ϵ.$$

For each $k\in M$ and each $W\in 𝒩(e)$, we have $f(a_k)W\cap U\ne \mathrm{\varnothing }$ and $f(a_k)W\cap (G\setminus \overline{U})\ne \mathrm{\varnothing }$. Take $y_k\in W$ such that $f(a_k)y_k\notin \overline{U}$, where $k\in M$. Choosing $g\in G^{\bullet }$ with $P(g)=P(f)$ and $g(a_k)=y_k$ for each $k\in M$, and $g(r)=e$ otherwise, we have that $g\in O(W,\delta )$ for all $\delta >0$ and

$$\mu (\{r\in J:(fg)(r)\notin \overline{U}\})=1.$$

Thus $fg\in fO(W,\delta )⊈O(\overline{U},ϵ)$, that is, $f\notin \mathrm{Int}O(\overline{U},ϵ)$. Hence, $\mathrm{Int}\overline{O(U,ϵ)}\subset O(\mathrm{Int}\overline{U},ϵ)$. This completes the proof.

The subsequent theorem is the main result of this section, it follows from the combination of Lemmas 3.1 and 3.3.

Theorem 3.5.

Let $G$ be a semitopological group. Then the local bases

$${({\beta }_{ro})}^{\bullet }=\{O(\mathrm{Int}\overline{U},ϵ):U\in 𝒩(e),ϵ>0\}$$

and

$${({\beta }^{\bullet })}_{ro}=\{\mathrm{Int}\overline{O(V,ϵ)}:V\in 𝒩(e),ϵ>0\}$$

at $e^{\bullet }$ generate the same topology on $G^{\bullet }$, that is, ${(G^{\bullet })}_{sr}$ and ${(G_{sr})}^{\bullet }$ are topologically isomorphic.

In [7], M. Tkachenko proved that in paratopological groups, the $T_3$-reflection and semiregularization coincide. Therefore, combining Theorems 3.5, 2.21 and [7, Theorem 3.8], we obtain the following result.

Theorem 3.6.

Let $G$ be a paratopological group. Then the groups $Reg(G^{\bullet })$ and $Reg{(G)}^{\bullet }$ are topologically isomorphic.

4. Conclusions

From this work we can conclude the following:

1. In the category of paratopological groups, the functors $T_i$ and ^∙ commute for $i\in \{0,1,2,3,3.5\}$.

2. In the category of semitopological groups, the functors $T_i$ and ^∙ commute for $i\in \{0,1,2\}$. Moreover, the semiregularization and the functor ^∙ also commute in this category.

5. Open Problems

Since the semiregularization of a semitopological group does not, in general, coincide with its $T_3$-reflection, it is natural to consider the following problems.

Problem 5.1.

Describe in internal terms the kernel of the canonical homomorphism ${\phi }_{G,3}$ of a semitopological group $G$ onto $T_3(G)$.

Problem 5.2.

Let $G$ be a semitopological group. Are the semitopological groups $T_3(G^{\bullet })$ and ${(T_3(G))}^{\bullet }$ topologically isomorphic?

We recall that a class of spaces is a $\mathrm{𝑃𝑆}$-class if it is closed under arbitrary products, taking subspaces, and contains a singleton.

Let $G$ be a topological group and $𝒞$ be a $\mathrm{𝑃𝑆}$-class. By [7, Theorem 2.3], the $𝒞$-reflection $𝒞(G)$ of $G$ exists and is a topological group.

Problem 5.3.

Is ${\left(𝒞(G)\right)}^{\bullet }$ an element of the $\mathrm{𝑃𝑆}$-class $𝒞$?

Problem 5.4.

Are the topological groups $𝒞(G^{\bullet })$ and ${\left(𝒞(G)\right)}^{\bullet }$ topologically isomorphic?