Applied General Topology

Dynamics of induced mappings on symmetric products, some answers

2022-04-29T11:53:00+02:00

Let X be a metric continuum and n a positive integer. Let F_n (X) be the hyperspace of nonempty subsets of X with at most n points. If 0 < m < n, we consider the quotient space Fⁿ_m (X) = F_n (X)/F_m (X). Given a mapping f from X into X, we consider the induced mappings f_n from F_n (X) into F_n (X) and fⁿ_m from Fⁿ_m (X) into Fⁿ_m (X). In this paper we study the relations among the dynamics of the mappings f, f_n, and fⁿ_m and we answer some questions, by F. Barragán, A. Santiago-Santos and J. Tenorio, related to the properties: minimality, irreducibility, strong transitive and turbulence.

A Urysohn lemma for regular spaces

2022-02-14T20:21:09+01:00

Using the concept of m-open sets, M-regularity and M-normality are introduced and investigated. Both these notions are closed under arbitrary product. M-normal spaces are found to satisfy a result similar to Urysohn lemma. It is shown that closed sets can be separated by m-continuous functions in a regular space.

Partial actions of groups on hyperspaces

2021-12-14T11:30:24+01:00

Let X be a compact Hausdorff space. In this work we translate partial actions of X to partial actions on some hyperspaces determined by X, this gives an endofunctor 2^- in the category of partial actions on compact Hausdorff spaces which generates a monad in this category. Moreover, structural relations between partial actions θ on X and partial determined by 2^θ as well as their corresponding globalizations are established.

On the group of homeomorphisms on R: A revisit

2021-11-28T02:42:15+01:00

In this article, we prove that the group of all increasing homeomorphisms on R has exactly five normal subgroups, and the group of all homeomorphisms on R has exactly four normal subgroups. There are several results known about the group of homeomorphisms on R and about the group of increasing homeomorphisms on R ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. In this paper, we analyze this structure in some detail.

The largest topological ring of functions endowed with the m-topology

2022-04-21T01:12:09+02:00

The purpose of this article is to identify the largest subring of the ring of all real valued functions on a Tychonoff space X, which forms a topological ring endowed with the m-topology.

C*-algebra valued quasi metric spaces and fixed point results with an application

2022-01-10T09:08:37+01:00

In this paper, we introduce the notion of C*-algebra valued quasi metric space to generalize the notion of C*-algebra valued metric space and investigate the topological properties besides proving some core fixed point results. Finally, we employ our one of the main results to examine the existence and uniqueness of the solution for a system of Fredholm integral equations.

Cardinal invariants and special maps of quasicontinuous functions with the topology of pointwise convergence

2022-01-24T09:37:16+01:00

For topological spaces X and Y, let Q_p(X,Y) be the space of all quasicontinuous functions from X to Y with the topology of pointwise convergence. In this paper, we study the cardinal invariants such as cellularity, character, weight, density, pseudocharacter and spread of the space Q_p(X,Y). We also discuss the properties of the restriction and induced maps related to the space Q_p(X,Y).

On set star-Lindelöf spaces

2022-04-12T18:21:10+02:00

A space X is said to be set star-Lindelöf if for each nonempty subset A of X and each collection U of open sets in X such that A ⊆⋃U, there is a countable subset V of U such that A ⊆ St (⋃V,U). The class of set star-Lindelöf spaces lie between the class of Lindel öf spaces and the class of star-Lindelöf spaces. In this paper, we investigate the relationship between set star-Lindelöf spaces and other related spaces by providing some suitable examples and study the topological properties of set star-Lindelöf spaces.

Results about S2-paracompactness

2022-08-11T13:18:30+02:00

We present new results regarding S2-paracompactness, that we estab-lished in [1], and its relation with other properties such as S-normality, epinormality and L-paracompactness.

Zariski topology on the spectrum of fuzzy classical primary submodules

2022-04-04T18:48:34+02:00

Let R be a commutative ring with identity and M a unitary R-module. The fuzzy classical primary spectrum F cp.spec(M) is the collection of all fuzzy classical primary submodules A of M, the recent generalization of fuzzy primary ideals and fuzzy classical prime submodules. In this paper, we topologize FM(M) with a topology having the fuzzy primary Zariski topology on the fuzzy classical primary spectrum F cp.spec(M) as a subspace topology, and investigate the properties of this topological space.

The Zariski topology on the graded primary spectrum of a graded module over a graded commutative ring

2022-01-26T15:20:36+01:00

Let R be a G-graded ring and M be a G-graded R-module. We define the graded primary spectrum of M, denoted by PSG(M), to be the set of all graded primary submodules Q of M such that (GrM(Q) :RM) = Gr((Q:RM)). In this paper, we define a topology on PSG(M) having the Zariski topology on the graded prime spectrum SpecG(M) as a subspace topology, and investigate several topological properties of this topological space.

Fixed point theorems for F- contraction mapping in complete rectangular M-metric space

2022-04-01T22:56:12+02:00

In this paper, we prove a fixed point result for F- contraction principle in the framework of rectangular M-metric space. An example is also adopted to exhibit the utility of our result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equation.

Fixed point theorems for a new class of nonexpansive mappings

2022-04-01T21:03:06+02:00

We consider a new class of nonlinear mappings that generalizes two well-known classes of nonexpansive type mappings and extends some other classes of mappings. We present some existence and convergence results for this class of mappings. Some illustrative examples presented herein show the generality of the obtained results.

Fixed point results of enriched interpolative Kannan type operators with applications

2021-11-24T16:25:51+01:00

The purpose of this paper is to introduce the class of enriched interpolative Kannan type operators on Banach space that contains the
classes of enriched Kannan operators, interpolative Kannan type contraction operators and some other classes of nonlinear operators. Some examples are presented to support the concepts introduced herein. A convergence theorem for the Krasnoselskij iteration method to approximate fixed point of the enriched interpolative Kannan type operators is proved. We study well-posedness, Ulam-Hyers stability and periodic point property of operators introduced herein. As an application of the main result, variational inequality problems is solved.

Best proximity point (pair) results via MNC in Busemann convex metric spaces

2021-12-15T21:33:02+01:00

In this paper, we present a new class of cyclic (noncyclic) α-ψ and β-ψ condensing operators and survey the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in the setting of reflexive Busemann convex spaces. Then an application of the main existence result to study the existence of an optimal solution for a system of differential equations is demonstrated.

Fredholm theory for demicompact linear relations

2022-01-11T16:18:08+01:00

We first attempt to determine conditions on a linear relation T such that μT becomes a demicompact linear relation for each μ ∈ [0,1)(see Theorems 2.4 and 2.5). Second, we display some results on Fredholm and upper semi-Fredholm linear relations involving a demicompact one(see Theorems 3.1 and 3.2). Finally, we provide some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3).

Remarks on fixed point assertions in digital topology, 5

2022-01-05T20:11:09+01:00

As in [6, 3, 4, 5], we discuss published assertions concerning fixed points in “digital metric spaces” - assertions that are incorrect or incorrectly proven, or reduce to triviality.

The ε-approximated complete invariance property

2021-11-11T12:37:16+01:00

In the present paper we introduce a generalization of the complete invariance property (CIP) for metric spaces, which we will call the
 ε-approximated complete invariance property (ε-ACIP). For our goals, we will use the so called degree of nondensifiability 
(DND) which, roughly speaking, measures (in the specified sense) the distance from a bounded metric space to its class of Peano continua. 
Our main result relates the ε-ACIP with the DND and, in particular, proves that a densifiable metric space has the ε-ACIP for each ε>0. 
Also, some essentials differences between the CIP and the ε-ACIP are shown.

Classical solutions for the Euler equations of compressible fluid dynamics: A new topological approach

2022-06-16T21:30:21+02:00

In this article we study a class of Euler equations of compressible fluid dynamics. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results.