Applied General Topology

Countable networks on Malykhin's maximal topological group

Edgar Márquez — 2023-10-02

We present a solution to the following problem: Does every countable and non-discrete topological (Abelian) group have a countable network with infinite elements? In fact, we show that no maximal topological space allows for a countable network with infinite elements. As a result, we answer the question in the negative. The article also focuses on Malykhin's maximal topological group constructed in 1975 and establishes some unusual properties of countable networks on this special group G. We show, in particular, that for every countable network N for G, the family of finite elements of N is also a network for G.

Fixed points results for various types of interpolative cyclic contraction

Mohamed Edraoui — 2023-10-02

In this paper, we introduce four new types of contractions called in this order Kannan-type cyclic contraction via interpolation, interpolative Ćirić-Reich-Rus type cyclic contraction, and we prove the existence and uniqueness for a fixed point for each situation.

★-quasi-pseudometrics on algebraic structures

Shi-Yao He — 2023-10-02

In this paper, we introduce some concepts of ★-(quasi)-pseudometric spaces, and give an example which shows that there is a ★-quasi-pseudometric space which is not a quasi-pseudometric space. We also study the conditions under which ★-quasi-pseudometric semitopological groups are paratopological groups or topological groups.

Ćirić-generalized contraction via wt−distance

Hosein Lakzian — 2023-10-02

In this present paper, besides other things, we introduce the concept of Ćirić-generalized contractions via wt−distance and then we will prove some new fixed point results for these mappings, which generalize and improve fixed point theorems by L. B. Ćirić in [9, 8, 10] and also, B. E. Rhoades in [23]. Some examples illustrate usefulness of the new results. At the end, we will give some applications to nonlinear fractional differential equations.

Remarks on fixed point assertions in digital topology, 6

Laurence Boxer — 2023-10-02

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

Fixed points of set-valued mappings in Menger probabilistic metric spaces endowed with an amorphous binary relation

Gopi Prasad — 2023-10-02

In this paper, we prove the existence of fixed point results for set-valued mappings in Menger probabilistic metric spaces equipped with an amorphous binary relation and a Hadžić -type t-norm. For the usability of such findings we present a Kelisky-Rivlin type result for a class of Bernstein type special operators introduced by Deo et. al. [Appl. Math. Comput. 201, (2008), 604-612 ] on the space C([ 0, n/n+1]). In this way, these investigations extend, modify and generalize some prominent recent fixed point results of the existing literature.

Partial actions on limit spaces

Bernd Losert — 2023-10-02

G-compactifications of continuous partial actions in the category of limit spaces are considered. In particular, sufficient conditions are given to ensure that (G, X, α) has a largest regular G-compactification.

Rough representations of rough topological groups

Essoyomewè Kieou — 2023-10-02

In this paper, the concept of rough representation of a rough topological group on a Banach space is explored. Mainly, the continuity and the irreducibility of rough representations are studied.

A note on the fixed point theorem of F-contraction mappings in rectangular M-metric space

Mujahid Abbas — 2023-10-02

In this note, we show that the main result (Theorem 3.2) due to Asim et al. (Appl. Gen. Topol., 23(2), 363-376 (2022) https://doi.org/10.4995/agt.2022.17418) is still valid if we remove the assumption of continuity of the mapping.

On graph induced symbolic systems

Prashant Kumar — 2023-10-02

In this paper, we investigate shift spaces arising from a multidimensional graph G. In particular, we investigate nonemptiness and existence of periodic points for a multidimensional shift space. We derive sufficient conditions under which these questions can be answered affirmatively. We investigate the structure of the shift space using the generating matrices. We prove that any d-dimensional shift of finite type is finite if and only if it is conjugate to a shift generated through permutation matrices. We prove that if any triangular pattern of the form a b c can be extended to a 1 x 1 square then the two dimensional shift space possesses periodic points. We introduce the notion of an E-pair for a two dimensional shift space. Using the notion of an E-pair, we derive sufficient conditions for non-emptiness of the two dimensional shift space under discussion.

Strong Fréchet properties of spaces constructed from squares and AD families

William Chen-Mertens — 2023-10-02

We answer questions of Arhangel'skiĭ using spaces defined from combinatorial objects. We first establish further convergence properties of a space constructed from □ ( κ ) showing it is Fréchet-Urysohn for finite sets and a w-space that is not a W-space. We also show that under additional assumptions it may be not bi-sequential, and so providing a consistent example of an absolutely Fréchet α₁ space that is not bisequential. In addition, if we do not require the space being α₁, we can construct a ZFC example of a countable absolutely Fréchet space that is not bisequential from an almost disjoint family of subsets of the natural numbers.

Some properties defined by relative versions of star-covering properties II

Maddalena Bonanzinga — 2023-10-02

In this paper we consider some recent relative versions of Menger property called set strongly star Menger and set star Menger properties and the corresponding Hurewicz-type properties. In particular, using [2], we "easily" prove that the set strong star Menger and set strong star Hurewicz properties are between countable compactness and the property of having countable extent. Also we show that the extent of a regular set star Menger or a set star Hurewicz space cannot exceed c. Moreover, we construct (1) a consistent example of a set star Menger (set star Hurewicz) space which is not set strongly star Menger (set strongly star Hurewicz) and show that (2) the product of a set star Menger (set star Hurewicz) space with a compact space need not be set star Menger (set star Hurewicz). In particular, (1) and (2) answer some questions posed by Kočinac, Konca and Singh in [17] and [23].

Smooth fans that are endpoint rigid

Rodrigo Hernández-Gutiérrez — 2023-10-02

Let X be a smooth fan and denote its set of endpoints by E(X). Let E be one of the following spaces: the natural numbers, the irrational numbers, or the product of the Cantor set with the natural numbers. We prove that there is a smooth fan X such that E(X) is homeomorphic to E and for every homeomorphism h : X → X , the restriction of h to E(X) is the identity. On the other hand, we also prove that if X is any smooth fan such that E(X) is homeomorphic to complete Erdős space, then X is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by Włodzimierz Charatonik.

Some topological and cardinal properties of the Nτφ-nucleus of a space X

F. G. Mukhamadiev — 2023-10-02

In this paper, we study the behavior of some topological and cardinal properties of topological spaces under the influence of the N_τ^φ -kernel of a space X. It has been proved that the N_τ^φ-kernel of a space X preserves the density and the network π - weight of normal spaces. Besides, shown that the N-compact kernel of a space X preserves the Souslin properties, the weight, the density, and the π -network weight of normal spaces.

On e-spaces and rings of real valued e-continuous functions

Susan Afrooz — 2023-10-02

Whenever the closure of an open set is also open, it is called e-open and if a space have a base consisting of e-open sets, it is called e-space. In this paper we first introduce and study e-spaces and e-continuous functions (we call a function f from a space X to a space Y an e-continuous at x ∈ X if for each open set V containing f(x) there is an e-open set containing x with f ( U ) ⊆ V ). We observe that the quasicomponent of each point in a space X is determined by e-continuous functions on X and it is characterized as the largest set containing the point on which every e-continuous function on X is constant. Next, we study the rings C_e( X ) of all real valued e-continuous functions on a space X. It turns out that C_e( X ) coincides with the ring of real valued clopen continuous functions on X which is a C(Y) for a zero-dimensional space Y whose elements are the quasicomponents of X. Using this fact we characterize the real maximal ideals of C_e( X ) and also give a natural representation of its maximal ideals. Finally we have shown that C_e( X ) determines the topology of X if and only if it is a zero-dimensional space.

Fixed point of Lipschitz type mappings

Ravindra K. Bisht — 2023-10-02

In this paper, we prove some fixed point theorems for Lipschitz type mappings in the setting of metric spaces. Our results open up the unexplored area of fixed points of Lipschitz type mappings for investigation.

Pettis property for Polish inverse semigroups

Karen Arana — 2023-10-02

We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections between subsets of N. We also study whether our examples satisfy automatic continuity.

On φ-contractions and fixed point results in fuzzy metric spaces

Morteza Saheli — 2023-10-02

In this paper, φ-contractions are defined and then, some new fixed point theorems are established for certain nonlinear mappings associated with one-dimensional (c)-comparison functions in fuzzy metric spaces. Next, generalized φ-contractions are defined by using five-dimensional (c)-comparison functions, and the existence of fixed points for nonlinear maps on fuzzy metric spaces is studied. Moreover, some examples are given to illustrate our results.