Applied General Topology - Vol 22, No 1 (2021)

Tabla de contenidos

A glance into the anatomy of monotonic maps
Metric topology on the moduli space
On soft quasi-pseudometric spaces
Further remarks on group-2-groupoids
Intermediate rings of complex-valued continuous functions
Equicontinuous local dendrite maps
Ideal spaces
Convexity and boundedness relaxation for fixed point theorems in modular spaces
From interpolative contractive mappings to generalized Ciric-quasi contraction mappings
Convexity and freezing sets in digital topology
Remarks on the rings of functions which have a finite numb er of di scontinuities
On sheaves of Abelian groups and universality
Metric spaces related to Abelian groups
Digital homotopic distance between digital functions
Duality of locally quasi-convex convergence groups
On the Menger and almost Menger properties in locales

URI permanente para esta colecciónhttps://riunet.upv.es/handle/10251/165208

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Access status: Abierto ,
On the Menger and almost Menger properties in locales
(Universitat Politècnica de València, 2021-04-01) Bayih, Tilahun; Dube, Themba; Ighedo, Oghenetega; National Research Foundation, South Africa
[EN] The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober TD-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.
Access status: Abierto ,
Duality of locally quasi-convex convergence groups
(Universitat Politècnica de València, 2021-04-01) Sharma, Pranav
[EN] In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be c-reflexive. Further, we prove that every character group of a convergence group is locally quasi-convex.
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Digital homotopic distance between digital functions
(Universitat Politècnica de València, 2021-04-01) Borat, Ayse
[EN] In this paper, we define digital homotopic distance and give its relation with LS category of a digital function and of a digital image. Moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance.
Access status: Abierto ,
Metric spaces related to Abelian groups
(Universitat Politècnica de València, 2021-04-01) Veisi, Amir; Delbaznasab, Ali
[EN] When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ clXA if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e < g ∈ G and d is bounded, then d′ (A, B) < g if and only if A ⊆ Nd(B, g) and B ⊆ Nd(A, g), where Nd(A, g) = {x ∈ X : d(x, A) < g}, dB(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{dA(B), dB(A)}.
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On sheaves of Abelian groups and universality
(Universitat Politècnica de València, 2021-04-01) Iliadis, S.D.; Sadovnichy, Yu. V.
[EN] Universal elements are one of the most essential parts in research fields, investigating if there exist (or not) universal elements in different classes of objects. For example, classes of spaces and frames have been studied under the prism of this universality property. In this paper, studying classes of sheaves of Abelian groups, we construct proper universal elements for these classes, giving a positive answer to the existence of such elements in these classes.
Access status: Abierto ,
Remarks on the rings of functions which have a finite numb er of di scontinuities
(Universitat Politècnica de València, 2021-04-01) Ahmadi Zand, Mohammad Reza; Khosravi, Zahra
[EN] Let X be an arbitrary topological space. F(X) denotes the set of all real-valued functions on X and C(X)F denotes the set of all f ∈ F(X) such that f is discontinuous at most on a finite set. It is proved that if r is a positive real number, then for any f ∈ C(X)F which is not a unit of C(X)F there exists g ∈ C(X)F such that g ≠ 1 and f = gr f. We show that every member of C(X)F is continuous on a dense open subset of X if and only if every non-isolated point of X is nowhere dense. It is shown that C(X)F is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein.
Access status: Abierto ,
Convexity and freezing sets in digital topology
(Universitat Politècnica de València, 2021-04-01) Boxer, Laurence
[EN] We continue the study of freezing sets in digital topology, introduced in [4]. We show how to find a minimal freezing set for a "thick" convex disk X in the digital plane Z^2. We give examples showing the significance of the assumption that X is convex.
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From interpolative contractive mappings to generalized Ciric-quasi contraction mappings
(Universitat Politècnica de València, 2021-04-01) Roy, Kushal; Panja, Sayantan; Council of Scientific and Industrial Research, India
[EN] In this article we consider a restricted version of Ciric-quasi contraction mapping for showing that this mapping generalizes several known interpolative type contractive mappings. Also here we introduce the concept of interpolative strictly contractive type mapping T and prove a fixed point theorem for such mapping over a T-orbitally compact metric space. Some examples are given in support of our established results. Finally we give an observation regarding (λ, α, β)-interpolative Kannan contractions introduced by Gaba et al.
Access status: Abierto ,
Convexity and boundedness relaxation for fixed point theorems in modular spaces
(Universitat Politècnica de València, 2021-04-01) Lael, Fatemeh; Shabanian, Samira
[EN] Although fixed point theorems in modular spaces have remarkably applied to a wide variety of mathematical problems, these theorems strongly depend on some assumptions which often do not hold in practice or can lead to their reformulations as particular problems in normed vector spaces. A recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further. In this paper, we focus on convexity and boundedness of modulars in fixed point results taken from the literature for contractive correspondence and single-valued mappings. To relax these two assumptions, we seek to identify the ties between modular and b-metric spaces. Afterwards we present an application to a particular form of integral inclusions to support our generalized version of Nadler’s theorem in modular spaces.
Access status: Abierto ,
Ideal spaces
(Universitat Politècnica de València, 2021-04-01) Mitra, Biswajit; Chowdhury, Debojyoti
[EN] Let C∞ (X) denote the family of real-valued continuous functions which vanish at infinity in the sense that {x ∈ X : |f(x)| ≥ 1/n} is compact in X for all n ∈ N. It is not in general true that C∞ (X) is an ideal of C(X). We define those spaces X to be ideal space where C∞ (X) is an ideal of C(X). We have proved that nearly pseudocompact spaces are ideal spaces. For the converse, we introduced a property called “RCC” property and showed that an ideal space X is nearly pseudocompact if and only if X satisfies ”RCC” property. We further discussed some topological properties of ideal spaces.
Access status: Abierto ,
Equicontinuous local dendrite maps
(Universitat Politècnica de València, 2021-04-01) Salem, Aymen Haj; Hattab, Hawete; Rejeiba, Tarek
[EN] Let X be a local dendrite, and f : X → X be a map. Denote by E(X) the set of endpoints of X. We show that if E(X) is countable, then the following are equivalent:(1) f is equicontinuous;(2) fn (X) = R(f);(3) f| fn (X) is equicontinuous;(4) f| fn (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ;(5) ω(x, f) = Ω(x, f) for all x ∈ X.This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].
Access status: Abierto ,
Intermediate rings of complex-valued continuous functions
(Universitat Politècnica de València, 2021-04-01) Acharyya, Amrita; Acharyya, Sudip Kumar; Bag, Sagarmoy; Sack, Joshua
[EN] For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on X, let C ∗ (X, C) be its subring of bounded functions, and let Σ(X, C) denote the collection of all the rings that lie between C ∗ (X, C) and C(X, C). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in the rings P(X, C) in Σ(X, C) and in their real-valued counterparts P(X, C) ∩ C(X). These correlations culminate to the fact that the structure space of any such P(X, C) is βX. For any ideal I in C(X, C), we observe that C ∗ (X, C)+I is a member of Σ(X, C), which is further isomorphic to a ring of the type C(Y, C). Incidentally these are the only C-type intermediate rings in Σ(X, C) if and only if X is pseudocompact. We show that for any maximal ideal M in C(X, C), C(X, C)/M is an algebraically closed field, which is furthermore the algebraic closure of C(X)/M ∩C(X). We give a necessary and sufficient condition for the ideal CP (X, C) of C(X, C), which consists of all those functions whose support lie on an ideal P of closed sets in X, to be a prime ideal, and we examine a few special cases thereafter. At the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a P(X, C) in Σ(X, C).
Access status: Abierto ,
Further remarks on group-2-groupoids
(Universitat Politècnica de València, 2021-04-01) Temel, Sedat
[EN] The aim of this paper is to obtain a group-2-groupoid as a 2-groupoid object in the category of groups and also as a special kind of an internal category in the category of group-groupoids. Corresponding group-2-groupoids, we obtain some categorical structures related to crossed modules and group-groupoids and prove categorical equivalences between them. These results enable us to obtain 2-dimensional notions of group-groupoids.
Access status: Abierto ,
On soft quasi-pseudometric spaces
(Universitat Politècnica de València, 2021-04-01) Sabao, Hope; Otafudu, Olivier Olela
[EN] In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set.
Access status: Abierto ,
Metric topology on the moduli space
(Universitat Politècnica de València, 2021-04-01) Deng, Jialong
[EN] We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.
Access status: Abierto ,
A glance into the anatomy of monotonic maps
(Universitat Politècnica de València, 2021-04-01) Buzyakova, Raushan
[EN] Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.

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