Editorial

3 pages, 176 KiB

Open AccessEditorial

Advances in General Topology and Its Application

by Ljubiša D. R. Kočinac

Axioms 2023, 12(6), 579; https://doi.org/10.3390/axioms12060579 - 11 Jun 2023

Viewed by 1109

In recent years, mathematical and, in particular, topological models and methods have been used extensively in real-world problems related to economics, engineering, biology, computer science, medical science, social science, etc [...] Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

Research

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12 pages, 323 KiB

Open AccessArticle

A Note on Finite Coarse Shape Groups

by Ivan Jelić and Nikola Koceić-Bilan

Axioms 2023, 12(4), 377; https://doi.org/10.3390/axioms12040377 - 14 Apr 2023

Cited by 1 | Viewed by 754

Abstract

In this paper, we investigate properties concerning some recently introduced finite coarse shape invariants—the k-th finite coarse shape group of a pointed topological space and the k-th relative finite coarse shape group of a pointed topological pair. We define the notion [...] Read more.

In this paper, we investigate properties concerning some recently introduced finite coarse shape invariants—the k-th finite coarse shape group of a pointed topological space and the k-th relative finite coarse shape group of a pointed topological pair. We define the notion of finite coarse shape group sequence of a pointed topological pair

(X, X_{0}, x_{0})

as an analogue of homotopy and (coarse) shape group sequences and show that for any pointed topological pair, the corresponding finite coarse shape group sequence is a chain. On the other hand, we construct an example of a pointed pair of metric continua whose finite coarse shape group sequence fails to be exact. Finally, using the aforementioned pair of metric continua together with a pointed dyadic solenoid, we show that finite coarse-shape groups, in general, differ from both shape and coarse-shape groups. Full article

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19 pages, 347 KiB

Open AccessArticle

On Semi-Continuous and Clisquish Functions in Generalized Topological Spaces

by Elvis Aponte, Vadakasi Subramanian, Jhixon Macías and Muthumari Krishnan

Axioms 2023, 12(2), 130; https://doi.org/10.3390/axioms12020130 - 28 Jan 2023

Cited by 2 | Viewed by 995

Abstract

In this paper, we will focus on three types of functions in a generalized topological space, namely; lower and upper semi-continuous functions, and cliquish functions. We give some results for nowhere dense sets and for second category sets. Further, we discuss the nature [...] Read more.

In this paper, we will focus on three types of functions in a generalized topological space, namely; lower and upper semi-continuous functions, and cliquish functions. We give some results for nowhere dense sets and for second category sets. Further, we discuss the nature of cliquish functions in generalized metric spaces and provide the characterization theorem for cliquish functions in terms of nowhere dense sets. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

11 pages, 302 KiB

Open AccessArticle

An Application of wt-Distances to Characterize Complete b-Metric Spaces

by Salvador Romaguera

Axioms 2023, 12(2), 121; https://doi.org/10.3390/axioms12020121 - 26 Jan 2023

Cited by 2 | Viewed by 855

Abstract

The notion of

w t

-distance, introduced by Hussain et al. provides a natural generalization to the b-metric framework of the well-known and fruitful concept of w-distance, initiated by Kada et al. Since then, several authors have obtained fixed point theorems [...] Read more.

The notion of

w t

-distance, introduced by Hussain et al. provides a natural generalization to the b-metric framework of the well-known and fruitful concept of w-distance, initiated by Kada et al. Since then, several authors have obtained fixed point theorems for complete b-metric spaces with the help of

w t

-distances. In this note, we generalize the b-metric version of the celebrated Matkowski fixed point theorem, stated by Czerwik, by replacing the involved b-metric with any

w t

-distance on the corresponding complete b-metric space. From this result, we derive characterizations of complete b-metric spaces that constitute full generalizations of both a prominent characterization of metric completeness due to Suzuki and Takahashi, and the classical characterization of metric completeness obtained by Hu. Full article

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14 pages, 334 KiB

Open AccessArticle

Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

by Susmit Bagchi

Axioms 2022, 11(9), 425; https://doi.org/10.3390/axioms11090425 - 24 Aug 2022

Cited by 1 | Viewed by 1228

Abstract

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to [...] Read more.

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonné measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonné measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined. Full article

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16 pages, 317 KiB

Open AccessArticle

Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces

by Shubham Yadav, Dhananjay Gopal, Parin Chaipunya and Juan Martínez-Moreno

Axioms 2022, 11(8), 383; https://doi.org/10.3390/axioms11080383 - 05 Aug 2022

Cited by 2 | Viewed by 1232

Abstract

A Kuratowski topology is a topology specified in terms of closed sets rather than open sets. Recently, the binary metric was introduced as a symmetric, distributive-lattice-ordered magma-valued function of two variables satisfying a “triangle inequality” and subsequently proved that every Kuratowski topology can [...] Read more.

A Kuratowski topology is a topology specified in terms of closed sets rather than open sets. Recently, the binary metric was introduced as a symmetric, distributive-lattice-ordered magma-valued function of two variables satisfying a “triangle inequality” and subsequently proved that every Kuratowski topology can be induced by such a binary metric. In this paper, we define the strong convergence of a sequence in a binary metric space and prove that strong convergence implies convergence. We state the conditions under which strong convergence is equivalent to convergence. We define a strongly Cauchy sequence and a strong complete binary metric space. Finally, we give the strong completion of all binary metric spaces with a countable indexing set. Full article

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9 pages, 291 KiB

Open AccessArticle

Some Cardinal and Geometric Properties of the Space of Permutation Degree

by Ljubiša D. R. Kočinac, Farkhod G. Mukhamadiev and Anvar K. Sadullaev

Axioms 2022, 11(6), 290; https://doi.org/10.3390/axioms11060290 - 14 Jun 2022

Cited by 10 | Viewed by 1591

Abstract

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space

{SP}^{2} (R, τ (A))

[...] Read more.

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space

{SP}^{2} (R, τ (A))

of permutation degree of the Hattori space is discussed. In particular, it is shown that the space

{SP}^{2} (R, τ_{S})

contains a closed discrete subset of cardinality

c

. Moreover, it is shown that the functor

{SP}_{G}^{n}

preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces

{SP}_{G}^{n} X

and

{SP}_{G}^{n} Y

are also homotopically equivalent. As a result, it has been proved that the functor

{SP}_{G}^{n}

is a covariant homotopy functor. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

12 pages, 290 KiB

Open AccessArticle

On Soft Generalized ω-Closed Sets and Soft T_1/2 Spaces in Soft Topological Spaces

by Samer Al Ghour

Axioms 2022, 11(5), 194; https://doi.org/10.3390/axioms11050194 - 21 Apr 2022

Cited by 14 | Viewed by 1669

Abstract

In this paper, we define a soft generalized

ω

-closed set, which is a generalization of both the soft

ω

-closed set and the soft generalized closed set. We show that the classes of generalized closed sets and generalized

ω

-closed sets coincide [...] Read more.

In this paper, we define a soft generalized

ω

-closed set, which is a generalization of both the soft

ω

-closed set and the soft generalized closed set. We show that the classes of generalized closed sets and generalized

ω

-closed sets coincide in soft anti-locally countable soft topological spaces. Additionally, in soft locally countable soft topological spaces, we show that every soft set is a soft generalized

ω

-closed set. Furthermore, we prove that the classes of soft generalized closed sets and soft generalized

ω

-closed sets coincide in the soft topological space

(X, τ_{ω}, A)

. In addition to these, we determine the behavior of soft generalized

ω

-closed sets relative to soft unions, soft intersections, soft subspaces, and generated soft topologies. Furthermore, we investigate soft images and soft inverse images of soft generalized closed sets and soft generalized

ω

-closed sets under soft continuous, soft closed soft transformations. Finally, we continue the study of soft

T_{1 / 2}

spaces, in which we obtain two characterizations of these soft spaces, and investigate their behavior with respect to soft subspaces, soft transformations, and generated soft topologies. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

18 pages, 340 KiB

Open AccessFeature PaperArticle

Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces

by Mircea Sofonea

Axioms 2022, 11(3), 136; https://doi.org/10.3390/axioms11030136 - 16 Mar 2022

Cited by 1 | Viewed by 1490

Abstract

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory [...] Read more.

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory of maximal monotone operators, fixed point arguments and the properties of the subdifferential, both in the sense of Clarke and in the sense of convex analysis. These results lay the background in the study of various classes of inequalities. We use them to prove existence, uniqueness and continuous dependence results for the solution of elliptic and history-dependent variational–hemivariational inequalities. We also present some iterative methods in solving these inequalities, together with various convergence results. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

11 pages, 281 KiB

Open AccessArticle

Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

by Samer Al Ghour

Axioms 2021, 10(4), 339; https://doi.org/10.3390/axioms10040339 - 10 Dec 2021

Cited by 3 | Viewed by 1755

Abstract

As a weaker form of

ω

-paracompactness, the notion of

σ

-

ω

-paracompactness is introduced. Furthermore, as a weaker form of

σ

-

ω

-paracompactness, the notion of feebly

ω

-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are [...] Read more.

As a weaker form of

ω

-paracompactness, the notion of

σ

-

ω

-paracompactness is introduced. Furthermore, as a weaker form of

σ

-

ω

-paracompactness, the notion of feebly

ω

-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly

ω

-paracompact. Furthermore, it is proven hereinthat countably

ω

-paracompact

σ

-

ω

-paracompact topological spaces are

ω

-paracompact. Furthermore, it is proven hereinthat

σ

-

ω

-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat

ω

-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if

A

is a locally finite closed covering of a topological space

(X, τ)

with each

A \in A

being

ω

-paracompact and normal, then

(X, τ)

is

ω

-paracompact and normal, and as a corollary, a sum theorem for

ω

-paracompact normal topological spaces follows. Moreover, three open questions are raised. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

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