Applied General Topology

Lightly chaotic dynamical systems

Annamaria Miranda — 2024-10-01

In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Several examples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f : X → X , where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated functional dynamical system is lightly chaotic.

Graded pseudo weakly prime spectrum of graded topological modules

Tamem Al-Shorman — 2024-10-01

In this study, we introduce graded pseudo weakly prime submodules of G-graded R-modules, which are an extension of graded weakly prime ideals over G-graded rings. On the graded spectrum of graded pseudo weakly prime submodules, we investigate the Zariski topology. Different aspects of this topological space are investigated, and they are linked to the algebraic properties of the G-graded R-modules under study.

T_0 functional Alexandroff topologies are partial metrizable

Homeira Pajoohesh — 2024-10-01

If f : X → X is a function, the associated functional Alexandroff topology on X is the topology whose closed sets are { A ⊆ X : f ( A ) ⊆ A } . We prove that every functional Alexandroff topology is pseudopartial metrizable and every T₀ functional Alexandroff topology is partial metrizable.

Mizoguchi-Takahashi local contractions to Feng-Liu contractions

Pallab Maiti — 2024-10-01

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established.

On maximal nowhere dense sublocales

Mbekezeli Nxumalo — 2024-10-01

The aim of this paper is to study some variants of nowhere dense sublocales called maximal nowhere dense and homogeneous maximal nowhere dense sublocales. These concepts were initially introduced by Veksler in classical topology. We give some general properties of these sublocales and further examine their relationship with both inaccessible sublocales and remote sublocales. It turns out that a locale has all of its non-void nowhere dense sublocales maximal nowhere dense precisely when all of its its non-void nowhere dense sublocales are inaccessible. We show that the Booleanization of a locale is inaccessible with respect to every dense and open sublocale. In connection to remote sublocales, we prove that, if the supplement of an open dense sublocale S is homogeneous maximal nowhere dense, then every S^#-remote sublocale is ^*-remote from S. Every open localic map that sends dense elements to dense elements preserves and reflects maximal nowhere dense sublocales. If such a localic map is further injective, then it sends homogeneous maximal nowhere dense sublocales back and forth.

Some results on weaker forms of starn-CCC, weakly Lindelöf and starn-DCCC spaces

Ricardo Cruz-Castillo — 2024-10-01

In this paper we provide some general results about topological spaces X which satisfy starⁿ- P , weakly starⁿ- P and almost starⁿ- P, for P∈ { κ ᴄᴄ , W κ L , ᴅ κ ᴄᴄ } , where κ is an infinite cardinal number.
The particular cases when κ = ω , P ∈ { ᴄᴄᴄ , weakly Lindelöf , ᴅ ᴄᴄᴄ } are obtained. Furthermore, for the same classes of spaces defined by such P, by applying Erdős-Radó's theorem and using the rank l-diagonal notion, we establish some cardinal inequalities.

k-spaces of non-domain-valued geometric points

Amartya Goswami — 2024-10-01

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a k-algebra and this new “k-space” becomes a generalization of the corresponding Zariski space. We prove that a k-space is T₀, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such k-spaces. We conclude with a question about the construction of a sheaf of k-spaces similar to affine schemes.

On the hyperspaces of meager and regular continua

Javier Camargo — 2024-10-01

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite.
In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that | D ( X ) | ∉ { 2,3,4,5,8 , 9 } h t ) for any continuum X. Also, we prove that D(X) is never homeomorphic to ℕ . On the other hand, given a point p ∈ X , we consider the meager composant and the filament composant of p in X, denoted by M^X_p and Fcs_X(p), respectively, and we study some relations between M^X_p and Fcs_X(p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs ( X ) = { Fcs_X ( p ) : p ∈ X } can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to an open problem. Also, we rise open problems.

On star Rothberger spaces modulo an ideal

Susmita Sarkar — 2024-10-01

In this article, we introduce the ideal star-Rothberger property by coupling the notion of a star operator to that of an ideal Rothberger space, after which some of its topological characteristics are analysed. By creating relationships between a numbers of topological features with structures similar to the ideal star-Rothberger space, we reinforce the concept. In order to illustrate the differences between a number of related topological properties, we also provide several counter examples. Certain preservation-related properties under subspaces and functions are investigated. Lastly we find a way to express ideal star-Rothberger space by means of families of closed sets by bringing some modifications to the SSI^I property.

Withdrawn

Applied General Topology — 2024-10-01

The editorial board has withdrawn this article since we have been warned that one of the authors' names has been added without his permission.

Maintaining ethical standards is a priority for our journal, and adding an author who did not contribute to the research or writing of the manuscript contravenes the conditions of the agreement of cession signed by the authors.

The Publisher apologizes for any inconvenience this may cause.

Existence of fixed points of large MR-Kannan contractions in Banach Spaces

Rizwan Anjum — 2024-10-01

The purpose of this paper is to introduce the class of large MR-Kannan contractions on Banach space that contains the classes of Kannan, enriched Kannan, large Kannan, MR-Kannan contractions and some other classes of nonlinear operators. Some examples are presented to support the concepts introduced herein. We prove the existence of a unique fixed point for such a class of operators in Banach spaces.

H-partial uniform spaces and their application in the compression of digital images

Satya Narayan Shukla — 2024-10-01

Fixed point theorem is very important tool in different branches of mathematics. In this paper, we introduce partial uniform spaces as a generalization of uniform spaces and metric spaces; and study some basic properties. Various examples support the theory. We prove fixed point theorems for H-partial uniform spaces, by using a mapping called E-distance function. Finally, we give the applications of these fixed point theorems to compress digital images.

Remarks on fixed point assertions in digital topology, 8

Laurence Boxer — 2024-10-01

This paper continues a series in which we study deficiencies in previously published works concerning fixed point assertions for digital images.

An interpolative class of two-Lipschitz mappings of composition type

Khaled Hamidi — 2024-10-01

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 characterizing the adjoint of a two-Lipschitz operator, we prove a Schauder type theorem on the compactness of the adjoint. We study the extension of two-Lipschitz operators from the 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 the cartesian product of two pointed metric spaces admits an extension with the same two-Lipschitz norm under some requirements on domaine spaces.

A generalization of strongly irreducible ideals with a view towards rings of continuous functions

Jamal Hashemi — 2024-10-01

In this note, we introduce and study the concept of a semi-strongly irreducible ideal, a natural generalization of a strongly irreducible ideal. We say an ideal I of a commutative ring R is semi-strongly irreducible if for ideals J and K of R, the inclusion J ∩ K ⊆ I implies that either J² ⊆ I or K² ⊆ I . After some general results, the article focuses on semi-strongly irreducible ideals in rings of continuous functions.

Butterfly points and hyperspace selections

Valentin Gutev — 2024-10-01

If f is a continuous selection for the Vietoris hyperspace ℱ(X) of the nonempty closed subsets of a space X, then the point f(X)∊ X is not as arbitrary as it might seem at first glance. In this paper, we will characterise these points by local properties at them. Briefly, we will show that p=f(X) is a strong butterfly point precisely when it has a countable clopen base in cl(U) for some open set U⊂ X\{p} with cl(U)=U ∪ {p}. Moreover, the same is valid when X is totally disconnected at p=f(X) and p is only assumed to be a butterfly point. This gives the complete affirmative solution to a question raised previously by the author. Finally, when p=f(X) lacks the above local base-like property, we will show that ℱ(X) has a continuous selection h with the stronger property that h(S)=p for every closed S⊂X with p∈S.

On traditional Menger and Rothberger variations

Christopher Caruvana — 2024-10-01

We present a comprehensive report on the relationships between variations of the Menger and Rothberger selection properties with respect to w-covers and k-covers in the most general topological setting and address the finite productivity of some of these properties. We collect various examples that separate certain properties and we carefully identify which separation axioms simplify aspects of these properties. We finish with a consolidated list of open questions focused on topological examples.