Applied General Topology

Some classes of topological spaces related to zero-sets

F. Golrizkhatami — 2022-04-01

An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briefly CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z^#-embedded subspace of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, cl_TZ is a zero-set in T). In 6P.5 of [8] it was shown that a closed countable union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. cozero complemented)-space and other classes of topological spaces near to them. Some algebraic and topological equivalent conditions of CZ (resp. cozero complemented)-space are characterized. Examples are provided to illustrate and delimit our results.

Representations of bornologies

Homeira Pajoohesh — 2022-04-01

Bornologies abstract the properties of bounded sets of a metric space. But there are unbounded bornologies on a metric space like $\mathcal{P}(\RR)$ with the Euclidean metric.
We show that by replacing $[0,\infty)$ with a partially ordered monoid every bornology is the set of bounded subsets of a generalized metric mapped into a partially ordered monoid. We also prove that the set of bornologies on a set is the join completion of the equivalence classes of a relation on the power set of the set.

Some fixed point results for enriched nonexpansive type mappings in Banach spaces

Rahul Shukla — 2022-04-01

In this paper, we introduce two new classes of nonlinear mappings and present some new existence and convergence theorems for these mappings in Banach spaces. More precisely, we employ the Krasnosel'skii iterative method to obtain fixed points of Suzuki-enriched nonexpansive mappings under different conditions. Moreover, we approximate the fixed point of enriched-quasinonexpansive mappings via Ishikawa iterative method.

Investigation of topological spaces using relators

Gergely Pataki — 2022-04-01

In this paper, we define uniformities and topologies as relators and show the equivalences of these definitions with the classical ones. For this, we summarize the essential properties of relators, using their theory from earlier works of Á. Száz.
Moreover, we prove implications between important topological properties of relators and disprove others. Finally, we show that our earlier analogous definition [G. Pataki, Investigation of proximal spaces using relators, Axioms 10, no. 3 (2021): 143.] for uniformly and proximally filtered property is equivalent to the topological one.

At the end of this paper, uniformities and topologies are defined in the same way. This will give us new possibilities to compare these and other topological structures.

On certain new notion of order Cauchy sequences, continuity in (l)-group

Sudip Kumar Pal — 2022-04-01

In this paper, we introduce the notions of order quasi-Cauchy sequences, downward and upward order quasi-Cauchy sequences, order half Cauchy sequences. Next we consider an associated idea of continuity namely, ward order continuous functions [2] and investigate certain interesting results. The entire investigation is performed in (l)-group setting to extend the recent results in [5, 6].

Beyond the Hausdorff metric in digital topology

Laurence Boxer — 2022-04-01

Two objects may be close in the Hausdorff metric, yet have very different geometric and topological properties. We examine other methods of comparing digital images such that objects close in each of these measures have some similar geometric or topological property. Such measures may be combined with the Hausdorff metric to yield a metric in which close images are similar with respect to multiple properties.

Closed ideals in the functionally countable subalgebra of C(X)

Amir Veisi — 2022-04-01

In this paper, closed ideals in C_c(X), the functionally countable subalgebra of C(X), with the m_c-topology, is studied. We show that if
X is CUC-space, then C^*_c(X) with the uniform norm-topology is a Banach algebra. Closed ideals in C_c(X) as a modified countable analogue of closed ideals in C(X) with the m-topology are characterized. For a zero-dimensional space X, we show that a proper ideal in C_c(X) is closed if and only if it is an intersection of maximal ideals of C_c(X). It is also shown that every ideal in C_c(X) with the m_c-topology is closed if and only if X is a P-space if and only if every ideal in C(X) with the m-topology is closed. Moreover, for a strongly zero-dimensional space X, it is proved that a properly closed ideal in C^*_c(X) is an intersection of maximal ideals of C^*_c(X) if and only if X is pseudo compact. Finally, we show that if X is a P-space, then the family of e_c-ultrafilters and z_c-ultrafilter coincide.

On w-Isbell-convexity

Olivier Olela Otafudu — 2022-04-01

Chistyakov introduced and developed a concept of modular metric for an arbitrary set in order to generalise the classical notion of modular on a linear space. In this article, we introduce the theory of hyperconvexity in the setting of modular pseudometric that is herein called w-Isbell-convexity. We show that on a modular set, w-Isbell-convexity is equivalent to hyperconvexity whenever the modular pseudometric is continuous from the right on the set of positive numbers.

Fixed point index computations for multivalued mapping and application to the problem of positive eigenvalues in ordered space

Vo Viet Tri — 2022-04-01

In this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue.

Fixed point results with respect to a wt-distance in partially ordered b-metric spaces and its application to nonlinear fourth-order differential equation

Reza Babaei — 2022-04-01

In this paper we study the existence of the fixed points for Hardy-Rogers type mappings with respect to a wt-distance in partially ordered metric spaces. Our results provide a more general statement, since we replace a w-distance with a wt-distance and ordered metric spaces with ordered b-metric spaces. Some examples are presented to validate our obtained results and an application to nonlinear fourth-order differential equation are given to support the main results.

Topological transitivity of the normalized maps induced by linear operators

Pabitra Narayan Mandal — 2022-04-01

In this article, we provide a simple geometric proof of the following fact: The existence of transitive normalized maps induced by linear operators is possible only when the underlying space's real dimension is either 1 or 2 or infinity. A similar result holds for projective transformation as well.

Common new fixed point results on b-cone Banach spaces over Banach algebras

Hojjat Afshari — 2022-04-01

Recently Zhu and Zhai studied the concepts of cone b-norm and cone b-Banach space as generalizations of cone b-metric spaces and they

gave a definition of ϕ-operator and obtained some new fixed point theorems in cone b-Banach spaces over Banach algebras by using

ϕ-operator. In this paper we propose a notion of quasi-cone over Banach algebras, then by utilizing some new conditions and

following their work with introducing two mappings $\mathcal{T}$ and $\mathcal{S}$ we improve the fixed point theorems to the common

fixed point theorems. An example is given to illustrate the usability of the obtained results.

Boyd-Wong contractions in F-metric spaces and applications

Ashis Bera — 2022-04-01

The main aim of this paper is to study the Boyd-Wong type fixed point result in the F-metric context and apply it to obtain some existence and uniqueness criteria of solution(s) to a second order initial value problem and a Caputo fractional differential equation. We substantiate our obtained result by finding a suitable non-trivial example.

Some generalizations for mixed multivalued mappings

Mustafa Aslantaş — 2022-04-01

In this paper, we first introduce a new concept of KW-type m-contraction mapping. Then, we obtain some fixed point results for these mappings on M-metric spaces. Thus, we extend many well-known results for both single valued mappings and multivalued mappings such as the main results of Klim and Wardowski [13] and Altun et al. [4]. Also, we provide an interesting example to show the effectiveness of our result.

Topologically mixing extensions of endomorphisms on Polish groups

John Burke — 2022-04-01

In this paper we study the dynamics of continuous endomorphisms on Polish groups. We offer necessary and sufficient conditions for a continuous endomorphism on a Polish group to be weakly mixing. We prove that any continuous endomorphism of an abelian Polish group can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said group.

Selection principles: s-Menger and s-Rothberger-bounded groups

Muhammad Asad Iqbal — 2022-04-01

In this paper, selection principles are defined and studied in the realm of irresolute topological groups. Especially, s-Menger-bounded and s-Rothberger-bounded type covering properties are introduced and studied.

Topological Krasner hyperrings with special emphasis on isomorphism theorems

Manooranjan Singha — 2022-04-01

Krasner hyperring is studied in topological flavor. It is seen that Krasner hyperring endowed with topology, when the topology is compatible with the hyperoperations in some sense, fruits new results comprising algebraic as well as topological properties such as topological isomorphism theorems.

Numerical reckoning fixed points via new faster iteration process

Kifayat Ullah — 2022-04-01

In this paper, we propose a new iteration process which is faster than the leading S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61-79], Thakur et al. [App. Math. Comp. 275 (2016), 147-155] and M [Filomat 32, no. 1 (2018), 187-196] iterations for numerical reckoning fixed points. Using new iteration process, some fixed point convergence results for generalized α-nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. At the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes.

Alexandroff duplicate and βκ

Andrzej A Szymanski — 2022-04-01

We discuss spaces and the Alexandroff duplicates of those spaces that admit a Č-S embedding into the Čech-Stone compactification of a discrete space.