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A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 July 2024) | Viewed by 32289

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Prof. Dr. Salvador Romaguera

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Instituto Universitario de Matemática Pura y Aplicada-IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: general topology; fixed point theory; fuzzy metric spaces and related structures
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Dimitrios Georgiou

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Department of Mathematics, University of Patras, 265 00 Patras, Greece
Interests: topology; dimension theory; lattice theory; mathematical analysis; discrete mathematics

Prof. Dr. Manuel Sanchis

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Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Av. Vicent Sos Baynat s/n, C.P. 12071 Castelló de la Plana, Spain
Interests: mathematics; fuzzy set
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

General Topology constitutes an important mathematical branch not only for its ability to develop its own disciplinary body but also largely due to its basic nature, for its capacity for interaction and application in other fields of both mathematics and other scientific areas.

The main purpose of this issue is the publication of high-quality papers that gather new progress in any area of general topology (MSC 54) as well as in those where the use of methods and techniques from it is significant. Furthermore, non-elementary applications are also welcome. Of course, the keyword list below is not exhaustive.

Prof. Dr. Salvador Romaguera
Prof. Dr. Dimitrios Georgiou
Prof. Dr. Manuel Sanchis
Guest Editors

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Keywords

Metric and generalized metric spaces, and normed spaces
Fuzzy, soft, and probabilistic metric spaces, and fuzzy topology
Fixed point theory
Convergence in general topology
Categorical methods in general topology
Compactness and completeness
Extension of spaces
Uniform structures, proximity structures, and generalizations
Ordered topological spaces
Function spaces and hyperspaces
Set theoretic topology
Selection theory
Topological algebra
Topological dynamics
Methods of general topology in mathematical analysis
Methods of general topology in computer science
Applications

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Published Papers (18 papers)

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8 pages, 248 KiB

Open AccessArticle

Continuous Selections and Extremally Disconnected Spaces

by Adolfo Pimienta and Manuel Sanchis

Mathematics 2023, 11(4), 791; https://doi.org/10.3390/math11040791 - 4 Feb 2023

Viewed by 1475

Abstract

This paper deals with extremally disconnected spaces and extremally disconnected P-spaces. A space X is said to be extremally disconnected if, for every open subset V of X, the closure of V in X is also an open set. P-spaces are spaces in which the intersection of countably many open sets is an open set. The authors present a new characterization of extremally disconnected spaces, and the extremally disconnected P-spaces, by means of selection theory. If X is either an extremally disconnected space or an extremally disconnected P-space, then the usual theorems of extension of real-valued continuous functions for a dense subset S of X can be deduced from our results. A corollary of our outcomes is that every nondiscrete space X of nonmeasurable cardinality has a dense subset S such that S is not C-embedded in X. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

8 pages, 269 KiB

Open AccessArticle

Endograph Metric and a Version of the Arzelà–Ascoli Theorem for Fuzzy Sets

by Juan J. Font, Sergio Macario and Manuel Sanchis

Mathematics 2023, 11(2), 260; https://doi.org/10.3390/math11020260 - 4 Jan 2023

Viewed by 1825

Abstract

In this paper, we provide several Arzelà–Ascoli-type results on the space of all continuous functions from a Tychonoff space X into the fuzzy sets of

R^{n}

, [...] Read more.

In this paper, we provide several Arzelà–Ascoli-type results on the space of all continuous functions from a Tychonoff space X into the fuzzy sets of

R^{n}

(F_{U S C B} (R^{n}), H_{e n d})

, which are upper semi-continuous and have bounded support endowed with the endograph metric. Namely, we obtain positive results when X is considered to be a

k_{r}

-space and

C (X, (F_{U S C B} (R^{n}), H_{e n d}))

is endowed with the compact open topology, as well as when we assume that X is pseudocompact and

C (X, (F_{U S C B} (R^{n}), H_{e n d}))

is equipped with the uniform topology. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

14 pages, 331 KiB

Open AccessArticle

Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces

by Samed Özkan, Samirah Alsulami, Tesnim Meryem Baran and Muhammad Qasim

Mathematics 2022, 10(24), 4819; https://doi.org/10.3390/math10244819 - 19 Dec 2022

Cited by 1 | Viewed by 1269

Abstract

In this paper, we characterize each of

T_{0}

T_{1}

, Pre-Hausdorff and Hausdorff separation properties for the category

L - GS

of quantale-valued gauge spaces and

L

-gauge morphisms. Moreover, we investigate how these concepts are related to each other [...] Read more.

In this paper, we characterize each of

T_{0}

T_{1}

, Pre-Hausdorff and Hausdorff separation properties for the category

L - GS

of quantale-valued gauge spaces and

L

-gauge morphisms. Moreover, we investigate how these concepts are related to each other in this category. We show that

T_{0}

T_{1}

and

T_{2}

are equivalent in the realm of Pre-Hausdorff quantale-valued gauge spaces. Finally, we compare our results with the ones in some other categories. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

17 pages, 374 KiB

Open AccessArticle

Constructing a Linearly Ordered Topological Space from a Fractal Structure: A Probabilistic Approach

by José Fulgencio Gálvez-Rodríguez and Miguel Ángel Sánchez-Granero

Mathematics 2022, 10(23), 4518; https://doi.org/10.3390/math10234518 - 30 Nov 2022

Viewed by 1623

Abstract

Recent studies have shown that it is possible to construct a probability measure from a fractal structure defined on a space. On the other hand, a theory on cumulative distribution functions from an order on a separable linearly ordered topological space has been developed. In this paper, we show how to define a linear order on a space with a fractal structure, so that these two theories can be used interchangeably in both topological contexts. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

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15 pages, 303 KiB

Open AccessArticle

Topologies of Bihyperbolic Numbers

by Ana Savić, Merve Bilgin, Soley Ersoy and Marija Paunović

Mathematics 2022, 10(22), 4224; https://doi.org/10.3390/math10224224 - 11 Nov 2022

Viewed by 1446

Abstract

In this paper, we establish a correlation between the bihyperbolic numbers set and the semi-Euclidean space. There are three different norms on the semi-Euclidean space that allow us to define three different hypersurfaces on semi-Euclidean space. Hence, we construct some topological structures on these hypersurfaces called norm e, s, and t topologies. On the other hand, we introduce hyperbolic e, s, and t topologies on the bihyperbolic numbers set. Moreover, by using the idempotent and spectral representations of the bihyperbolic numbers, we introduce new topologies on the bihyperbolic numbers set. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

27 pages, 395 KiB

Open AccessArticle

Best Proximity Point Theorems without Fuzzy P-Property for Several (ψ − ϕ)-Weak Contractions in Non-Archimedean Fuzzy Metric Spaces

by Mi Zhou, Naeem Saleem, Antonio Francisco Roldán López de Hierro and Xiaolan Liu

Mathematics 2022, 10(21), 4031; https://doi.org/10.3390/math10214031 - 30 Oct 2022

Cited by 2 | Viewed by 1387

Abstract

This paper addresses a problem of global optimization in a non-Archimedean fuzzy metric space context without fuzzy P-property. Specifically, it concerns the determination of the fuzzy distance between two subsets of a non-Archimedean fuzzy metric space. Our approach to solving this problem is to find an optimal approximate solution to a fixed point equation. This approach has been well studied within a category of problems called proximity point problems. We explore some new types of

(ψ - ϕ)

-weak proximal contractions and investigate the existence of the unique best proximity point for such kinds of mappings. Subsequently, some fixed point results for corresponding contractions are proved, and some illustrative examples are presented to support the validity of the main results. Moreover, an interesting application in computer science, particularly in the domain of words has been provided. Our work is a fuzzy generalization of the proximity point problem by means of fuzzy fixed point method. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

13 pages, 312 KiB

Open AccessArticle

Basic Contractions of Suzuki-Type on Quasi-Metric Spaces and Fixed Point Results

by Salvador Romaguera

Mathematics 2022, 10(21), 3931; https://doi.org/10.3390/math10213931 - 23 Oct 2022

Cited by 6 | Viewed by 1701

Abstract

This paper deals with the question of achieving a suitable extension of the notion of Suzuki-type contraction to the framework of quasi-metric spaces, which allows us to obtain reasonable fixed point theorems in the quasi-metric context. This question has no an easy answer; in fact, we here present an example of a self map of Smyth complete quasi-metric space (a very strong kind of quasi-metric completeness) that fulfills a simple and natural contraction of Suzuki-type but does not have fixed points. Despite it, we implement an approach to obtain two fixed point results, whose validity is supported with several examples. Finally, we present a general method to construct non-

T_{1}

quasi-metric spaces in such a way that it is possible to systematically generate non-Banach contractions which are of Suzuki-type. Thus, we can apply our results to deduce the existence and uniqueness of solution for some kinds of functional equations which is exemplified with a distinguished case. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

23 pages, 366 KiB

Open AccessArticle

Representation of Lipschitz Maps and Metric Coordinate Systems

by Roger Arnau, Jose M. Calabuig and Enrique A. Sánchez Pérez

Mathematics 2022, 10(20), 3867; https://doi.org/10.3390/math10203867 - 18 Oct 2022

Cited by 1 | Viewed by 1580

Abstract

Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze “Pietsch Theorem inspired factorizations" through subspaces of

ℓ_{\infty}

and

L^{1}

, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane–Whitney formulas, are also given. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

13 pages, 315 KiB

Open AccessArticle

Interpolative Meir–Keeler Mappings in Modular Metric Spaces

by Erdal Karapınar, Andreea Fulga and Seher Sultan Yeşilkaya

Mathematics 2022, 10(16), 2986; https://doi.org/10.3390/math10162986 - 18 Aug 2022

Cited by 9 | Viewed by 1534

Abstract

Modular metric space is one of the most interesting spaces in the framework of the metric fixed point theory. The main goal of the paper is to provide some certain fixed point results in the context of modular metric spaces and non-Archimedean modular metric spaces. In particular, we examine the existence of interpolative Meir–Keeler contraction types via admissible mappings for fixed point theory. Our results bring together several results available in the current corresponding literature. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

10 pages, 300 KiB

Open AccessArticle

On Principal Fuzzy Metric Spaces

by Valentín Gregori, Juan-José Miñana, Samuel Morillas and Almanzor Sapena

Mathematics 2022, 10(16), 2860; https://doi.org/10.3390/math10162860 - 11 Aug 2022

Cited by 4 | Viewed by 1874

Abstract

In this paper, we deal with the notion of fuzzy metric space

(X, M, *)

, or simply

X

, due to George and Veeramani. It is well known that such fuzzy metric spaces, in general, are not completable [...] Read more.

In this paper, we deal with the notion of fuzzy metric space

(X, M, *)

, or simply

X

, due to George and Veeramani. It is well known that such fuzzy metric spaces, in general, are not completable and also that there exist p-Cauchy sequences which are not Cauchy. We prove that if every p-Cauchy sequence in

X

is Cauchy, then

X

is principal, and we observe that the converse is false, in general. Hence, we introduce and study a stronger concept than principal, called strongly principal. Moreover,

X

is called weak p-complete if every p-Cauchy sequence is p-convergent. We prove that if

X

is strongly principal (or weak p-complete principal), then the family of p-Cauchy sequences agrees with the family of Cauchy sequences. Among other results related to completeness, we prove that every strongly principal fuzzy metric space where

M

is strong with respect to an integral (positive) t-norm ∗ admits completion. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

7 pages, 267 KiB

Open AccessArticle

Freezing Sets for Arbitrary Digital Dimension

by Laurence Boxer

Mathematics 2022, 10(13), 2291; https://doi.org/10.3390/math10132291 - 30 Jun 2022

Cited by 1 | Viewed by 1218

Abstract

The study of freezing sets is part of the theory of fixed points in digital topology. Most of the previous work on freezing sets is for digital images in the digital plane

Z^{2}

. In this paper, we show how to obtain [...] Read more.

The study of freezing sets is part of the theory of fixed points in digital topology. Most of the previous work on freezing sets is for digital images in the digital plane

Z^{2}

. In this paper, we show how to obtain freezing sets for digital images in

Z^{n}

for arbitrary n, using the

c_{1}

and

c_{n}

adjacencies. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

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22 pages, 408 KiB

Open AccessArticle

Sequential Completeness for ⊤-Quasi-Uniform Spaces and a Fixed Point Theorem

by Gunther Jäger

Mathematics 2022, 10(13), 2285; https://doi.org/10.3390/math10132285 - 30 Jun 2022

Cited by 3 | Viewed by 1395

Abstract

We define sequential completeness for ⊤-quasi-uniform spaces using Cauchy pair ⊤-sequences. We show that completeness implies sequential completeness and that for ⊤-uniform spaces with countable ⊤-uniform bases, completeness and sequential completeness are equivalent. As an illustration of the applicability of the concept, we give a fixed point theorem for certain contractive self-mappings in a ⊤-uniform space. This result yields, as a special case, a fixed point theorem for probabilistic metric spaces. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

12 pages, 276 KiB

Open AccessArticle

Some Fixed-Point Theorems in Proximity Spaces with Applications

by Muhammad Qasim, Hind Alamri, Ishak Altun and Nawab Hussain

Mathematics 2022, 10(10), 1724; https://doi.org/10.3390/math10101724 - 18 May 2022

Cited by 3 | Viewed by 1564

Abstract

Considering the

ω

-distance function defined by Kostić in proximity space, we prove the Matkowski and Boyd–Wong fixed-point theorems in proximity space using

ω

-distance, and provide some examples to explain the novelty of our work. Moreover, we characterize Edelstein-type fixed-point theorem in [...] Read more.

Considering the

ω

-distance function defined by Kostić in proximity space, we prove the Matkowski and Boyd–Wong fixed-point theorems in proximity space using

ω

-distance, and provide some examples to explain the novelty of our work. Moreover, we characterize Edelstein-type fixed-point theorem in compact proximity space. Finally, we investigate an existence and uniqueness result for solution of a kind of second-order boundary value problem via obtained Matkowski-type fixed-point results under some suitable conditions. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

11 pages, 285 KiB

Open AccessArticle

The K_p,q-Compactness and K_p,q-Null Sequences, and the

K_{K_{p, q}}

-Approximation Property for Banach Spaces

by Ju Myung Kim

Mathematics 2022, 10(9), 1586; https://doi.org/10.3390/math10091586 - 7 May 2022

Cited by 1 | Viewed by 1772

Abstract

Let

K_{p, q}

(

1 \leq p, q \leq \infty

with

1 / p + 1 / q \geq 1

) be the ideal of

(p, q)

-compact operators. This paper investigates the compactness and null sequences [...] Read more.

Let

K_{p, q}

(

1 \leq p, q \leq \infty

with

1 / p + 1 / q \geq 1

) be the ideal of

(p, q)

-compact operators. This paper investigates the compactness and null sequences via

K_{p, q}

, and an approximation property of the ideal of

K_{p, q}

-compact operators. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

12 pages, 273 KiB

Open AccessArticle

Fixed Point Results for a New Rational Contraction in Double Controlled Metric-like Spaces

by Jia Deng, Xiaolan Liu, Yan Sun and Mi Zhou

Mathematics 2022, 10(9), 1439; https://doi.org/10.3390/math10091439 - 24 Apr 2022

Cited by 1 | Viewed by 1642

Abstract

In this paper, we present a new type of rational contraction in double controlled metric-like spaces and improve recent results of such spaces. Moreover, there is an example to verify the correctness of our results. Finally, we also obtain some new fixed point [...] Read more.

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

20 pages, 353 KiB

Open AccessArticle

Some Remarks on Strong Fuzzy Metrics and Strong Fuzzy Approximating Metrics with Applications in Word Combinatorics

by Raivis Bēts and Alexander Šostak

Mathematics 2022, 10(5), 738; https://doi.org/10.3390/math10050738 - 25 Feb 2022

Cited by 1 | Viewed by 1492

Abstract

Noticing that ordinary metrics do not present an adequate tool for the study of analytic problems of word combinatorics, as well as in the research of some problems related to theoretical computer science, we propose to use fuzzy metrics in this type of problems. Specifically, the so-called strong fuzzy metric seems to be more appropriate here. In the first part of the paper, we study some special classes of strong fuzzy metrics, topological and lattice properties of certain families of strong fuzzy metrics, and, more generally, strong k-fuzzy metrics. Noticing that one of the standard axioms of a strong fuzzy metric can be easily violated when applied in real situations, in the second part of the paper we introduce more general, approximating fuzzy metrics and illustrate their applicability with some numerical examples. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

20 pages, 324 KiB

Open AccessArticle

Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme

by Mujahid Abbas, Muhammad Waseem Asghar and Manuel De la Sen

Mathematics 2022, 10(2), 273; https://doi.org/10.3390/math10020273 - 16 Jan 2022

Cited by 14 | Viewed by 2233

Abstract

The aim of this paper is to propose a new faster iterative scheme (called

A A

-iteration) to approximate the fixed point of

(b, η)

-enriched contraction mapping in the framework of Banach spaces. It is also proved that our [...] Read more.

The aim of this paper is to propose a new faster iterative scheme (called

A A

-iteration) to approximate the fixed point of

(b, η)

-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using

A A

-iterative scheme. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

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Review

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30 pages, 435 KiB

Open AccessReview

Statistical and Ideal Convergences in Topology

by D. Georgiou, G. Prinos and F. Sereti

Mathematics 2023, 11(3), 663; https://doi.org/10.3390/math11030663 - 28 Jan 2023

Cited by 2 | Viewed by 1993

Abstract

The notion of convergence wins its own important part in the branch of Topology. Convergences in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, partially ordered sets (in short, posets), and fuzzy ordered sets (in short, fosets) develop significant chapters that attract the interest of many studies. In particular, statistical and ideal convergences play their own important role in all these areas. A lot of studies have been devoted to these special convergences, and many results have been proven. As a consequence, the necessity to produce and extend new results arises. Since there are many results on different kinds of convergences in different areas, we present a review paper on this research topic in order to collect the most essential results, which leads us to provide open questions for further investigation. More precisely, we present and gather definitions and results which have been proven for different kinds of convergences, mainly on statistical/ideal convergences, in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, posets, and fosets. Based on this presentation, we provide new open problems for further investigation on related topics. The study of these problems will create new “roads”, enriching the branch of convergences in the field of Topology. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

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