Applied General Topology

Expansive homeomorphisms on quasi-metric spaces

Olivier Olela Otafudu — 2024-04-02

The investigation of expansive homeomorphisms in metric spaces began with Utz in 1950. Thereafter, several authors have extensively studied this concept for different motivations. In this current article, we study expansive homeomorphism in the context of quasi-pseudometric spaces. This is motivated by the fact that any expansive homeomorphism on quasi-pseudometric space is again expansive homeomorphism on its induced pseudometric space but the converse is not true in general. Moreover, the study of orbit structures has been taken to consideration in this article. For instance, we investigate the denseness of orbits in the context of quasi-metric spaces.

Common fixed point theorems on complete and weak G-complete fuzzy metric spaces

Sugata Adhya — 2024-04-02

Motivated by Gopal and Vetro, we introduce a symmetric pair of β-admissible mappings and obtain common fixed point theorems for such a pair in complete and weak G-complete fuzzy metric spaces. In particular, we rectified, generalize and improve the common fixed point theorem obtained by Turkoglu and Sangurlu for two fuzzy ψ-contractive mappings. We include non-trivial examples to exhibit the generality and demonstrate our results.

On σ-compact Hattori spaces

Vitalij A. Chatyrko — 2024-04-02

 We present several characterizations of σ-compact Hattori spaces, and reject some possible characterization candidates of the spaces.

On some questions on selectively highly divergent spaces

Angelo Bella — 2024-04-02

A topological space X is selectively highly divergent (SHD) if for every sequence of non-empty open sets { U_n : n ∈ ω } of X, we can find x_n ∈ U_n such that the sequence {x_n : n ∈ ω } has no convergent subsequences. In this note we answer two questions related to this notion asked by Jiménez-Flores, Ríos-Herrejón, Rojas-Sánchez and Tovar-Acosta.

The extension of two-Lipschitz operators

Elhadj Dahia — 2024-04-02

The paper deals with some further results concerning the class of two-Lipschitz operators. We prove first an isometric isomorphism identification of two-Lipschitz operators and Lipschitz operators. After defining and characterize the adjoint of two-Lipschitz operator, we prove a Schauder type theorem on the compactness of the adjoint. We study the extension of two-Lipschitz operators from cartesian product of two complemented subspaces of a Banach space to the cartesian product of whole spaces. Also, we show that every two-Lipschitz functional defined on cartesian product of two pointed metric spaces admits an extension with the same two-Lipschitz norm, under some requirements on domaine spaces.

Homomorphisms of the lattice of slowly oscillating functions on the half-line

Yutaka Iwamoto — 2024-04-02

We study the space H(SO) of all homomorphisms of the vector lattice of all slowly oscillating functions on the half-line ℍ = [ 0 , ∞ ) . In contrast to the case of homomorphisms of uniformly continuous functions, it is shown that a homomorphism in H(SO) which maps the unit to zero must be zero-homomorphism. Consequently, we show that the space H(SO) without zero-homomorphism is homeomorphic to ℍ x (0, ∞). By describing a neighborhood base of zero-homomorphism, we show that H(SO) is homeomorphic to the space ℍ x (0, ∞) with one point added.

Fixed point approximations via generalized MR-Kannan mappings in Banach spaces

Ravindra K. Bisht — 2024-04-02

In this paper, we introduce a generalization of the concept of MR-Kannan type contractions and utilize this condition to derive new fixed point theorems under both contractive and non-contractive conditions. Our approach enhances various existing results related to enriched mappings.

Iterated function system of generalized cyclic F-contractive mappings

Talat Nazir — 2024-04-02

The aim of this paper is to study the sufficient conditions for the existence of attractor of a generalized cyclic iterated function system composed of a complete metric space and a finite collection of generalized cyclic F-contraction mappings. Some examples are presented to support our main results and concepts defined herein. The results proved in the paper extend and generalize various well known results in the existing literature.

Remarks on fixed point assertions in digital topology, 7

Laurence Boxer — 2024-04-02

This paper continues a series discussing flaws in published assertions concerning fixed points in digital images.

On interpolative Hardy-Rogers type cyclic contractions

Mohamed Edraoui — 2024-04-02

Recently, Karapınar introduced a new Hardy-Rogers type contractive mapping using the concept of interpolation and proved a fixed point theorem in complete metric space. This new type of mapping, called "interpolative Hardy-Rogers type contractive mapping" is a generalization of Hardy-Rogers's fixed point theorem. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for cyclic mappings on complete metric spaces. Moreover, an example is given to illustrate the usability of the obtained results.

Partial actions on quotient spaces and globalization

Luis Martínez — 2024-04-02

Given a partial action of a topological group G on a space X we determine properties P which can be extended from X to its globalization. We treat the cases when P is any of the following: Hausdorff, regular, metrizable, second countable, and having invariant metric. Further, for a normal subgroup H, we introduce and study a partial action of G/H on the orbit space of X; applications to invariant metrics and inverse limits are presented.

Partial actions of groups on profinite spaces

Luis Martínez — 2024-04-02

We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~_G is profinite, this leads to the fact that when G is profinite the enveloping space X_G is also profinite. Moreover, we provide conditions for the induced quotient map π_G : X → X / ∼_G of η to have a continuous section. Relations between continuous sections of π_G and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

Some existence and uniqueness results for a solution of a system of equations

Deepak Khantwal — 2024-04-02

This paper presents some existence and uniqueness results for a solution of a system of equations. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math.(Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results.

Hemi metric spaces and Banach fixed point theorems

Vildan Ozturk — 2024-04-02

In this work, we will define a new type metric with degree m and m+1 points which is called m-hemi metric as a generalization of two metric spaces. We will give and prove some topological properties. Also, Banach contraction mapping principle were proved and a application to Fredholm integral equation were gived in hemi metric spaces.

A study of new dimensions for ideal topological spaces

Fotini Sereti — 2024-04-02

In this paper new notions of dimensions for ideal topological spaces are inserted, called *-quasi covering dimension and ideal quasi
covering dimension. We study several of their properties and investigate their relations with types of covering dimensions like the *-covering dimension and the ideal covering dimension.

Compactness in the endograph uniformity

Iván Sánchez — 2024-04-02

Given a uniform space (X,U), we denote by F^*(X) to the family of fuzzy sets u in (X,U) such that u is normal and upper semicontinuous. Let U_E be the endograph uniformity on F^*(X). In this paper, we mainly characterize totally bounded and compact substes in the uniform space (F^*(X),U_E).

The degree of nondensifiability of linear bounded operators and its applications

Gonzalo García — 2024-04-02

In the present paper we define the degree of nondensifiability (DND for short) of a bounded linear operator T on a Banach space and analyze its properties and relations with the Hausdorff measure of non-compactness (MNC for short) of T. As an application of our results, we have obtained a formula to find the essential spectral radius of a bounded operator T on a Banach space as well as we have provided the best possible lower bound for the Hyers-Ulam stability constant of T in terms of the aforementioned DND.

On topological groups of monotonic automorphisms

Raushan Buzyakova — 2024-04-02

We study topological groups of monotonic automorphisms on a generalized ordered space L. We find a condition that is necessary and sufficient for the set of all monotonic automorphism on L along with the function composition and the topology of point-wise convergence to be a topological group.

Strongly Lipschitz (ℓp ,ℓq)-factorable mappings

Dahmane Achour — 2024-04-02

In this paper we study the space of strongly Lipschitz (ℓ_p ,ℓ_q) -factorable operators between metric spaces and a Banach spaces. In particular, a factorization of this class through ℓ_p and ℓ_q spaces is given. We show that this type of operators fits in the theory of composition α-Banach Lipschitz operator ideal. As a special case, we get a Lipschitz version of weakly p-nuclear operators.

Pointwise convergence on the rings of functions which are discontinuous on a set of measure zero

Dhananjoy Mandal — 2024-04-02

Consider the ring ℳ_∘ ( X , μ ) of functions which are discontinuous on a set of measure zero which is introduced and studied extensively in [2]. In this paper, we have introduced a ring B₁ ( X , μ ) of functions which are pointwise limits of sequences of functions in ℳ_∘ ( X , μ ) . We have studied various properties of zero sets, B₁ ( X , μ ) -separated and B₁ ( X , μ ) -embedded subsets of B₁ ( X , μ ) and also established an analogous version of Urysohn's extension theorem. We have investigated a connection between ideals of B₁ ( X , μ ) and Z_B -filters on X. We have studied an analogue of Gelfand-Kolmogoroff theorem in our setting. We have defined real maximal ideals of B₁ ( X , μ ) and established the result | ℛ M a x ( ℳ_∘ ( X , μ ) ) | = | ℛ M a x ( B₁ ( X , μ ) ) | , where ℛ M a x ( ℳ_∘ ( X , μ ) ) and ℛ M a x ( B₁ ( X , μ ) ) are the sets of all real maximal ideals of ℳ_∘ ( X , μ ) and B₁ ( X , μ ) respectively.