scholarly journals Closed subsets of compact-like topological spaces

2020 ◽  
Vol 21 (2) ◽  
pp. 201
Author(s):  
Serhii Bardyla ◽  
Alex Ravsky
Keyword(s):  
Topological Space ◽  
Topological Semigroup ◽  
Closed Subspace ◽  
Affirmative Answer ◽  
Topological Spaces ◽  
Hausdorff Topological Space ◽  
Bicyclic Monoid ◽  
Matrix Units

<p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of<br />countably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.</p>

scholarly journals On spaces defined by Pytkeev networks

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6115-6129 ◽  
Author(s):  
Xin Liu ◽  
Shou Lin
Keyword(s):  
Topological Space ◽  
Affirmative Answer ◽  
General Topology ◽  
Topological Spaces ◽  
Urysohn Space ◽  
Closed Mappings ◽  
The Relationship

The notions of networks and k-networks for topological spaces have played an important role in general topology. Pytkeev networks, strict Pytkeev networks and cn-networks for topological spaces are defined by T. Banakh, and S. Gabriyelyan and J. K?kol, respectively. In this paper, we discuss the relationship among certain Pytkeev networks, strict Pytkeev networks, cn-networks and k-networks in a topological space, and detect their operational properties. It is proved that every point-countable Pytkeev network for a topological space is a quasi-k-network, and every topological space with a point-countable cn-network is a meta-Lindel?f D-space, which give an affirmative answer to the following problem [25, 29]: Is every Fr?chet-Urysohn space with a pointcountable cs'-network a meta-Lindel?f space? Some mapping theorems on the spaces with certain Pytkeev networks are established and it is showed that (strict) Pytkeev networks are preserved by closed mappings and finite-to-one pseudo-open mappings, and cn-networks are preserved by pseudo-open mappings, in particular, spaces with a point-countable Pytkeev network are preserved by closed mappings.

scholarly journals On Fuzzy L-paracompact Topological Spaces

2021 ◽  
Vol 8 ◽  
pp. 38-40
Author(s):  
Francisco Gallego Lupiáñez
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Locally Compact ◽  
Hausdorff Topological Space ◽  
Characteristic Map ◽  
Covering Properties ◽  
Fuzzy Topological Space

The aim of this paper is to study fuzzy extensions of some covering properties defined by L. Kalantan as a modification of some kinds of paracompactness-type properties due to A.V.Arhangels'skii and studied later by other authors. In fact, we obtain that: if (X,T) is a topological space and A is a subset of X, then A is Lindelöf in (X,T) if and only if its characteristic map χ_{A} is a Lindelöf subset in (X,ω(T)). If (X,τ) is a fuzzy topological space, then, (X,τ) is fuzzy Lparacompact if and only if (X,ι(τ)) is L-paracompact, i.e. fuzzy L-paracompactness is a good extension of L-paracompactness. Fuzzy L₂-paracompactness is a good extension of L₂- paracompactness. Every fuzzy Hausdorff topological space (in the Srivastava, Lal and Srivastava' or in the Wagner and McLean' sense) which is fuzzy locally compact (in the Kudri and Wagner' sense) is fuzzy L₂-paracompact

scholarly journals Co-Compact Separation Axioms and Slight Co-Continuity

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1614
Author(s):  
Samer Al Ghour ◽  
Enas Moghrabi
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Closed Subspace ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Separation Axioms ◽  
Preservation Theorems

Via co-compact open sets we introduce co-T2 as a new topological property. We show that this class of topological spaces strictly contains the class of Hausdorff topological spaces. Using compact sets, we characterize co-T2 which forms a symmetry. We show that co-T2 propoerty is preserved by continuous closed injective functions. We show that a closed subspace of a co-T2 topological space is co-T2. We introduce co-regularity as a weaker form of regularity, s-regularity as a stronger form of regularity and co-normality as a weaker form of normality. We obtain several characterizations, implications, and examples regarding co-regularity, s-regularity and co-normality. Moreover, we give several preservation theorems under slightly coc-continuous functions.

scholarly journals Lacunary distributional convergence in topological spaces

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4297-4306
Author(s):  
Havva Uluçay ◽  
Mehmet Ünver
Keyword(s):  
Topological Space ◽  
Group Structure ◽  
Lacunary Sequence ◽  
Distribution Functions ◽  
Topological Spaces ◽  
Hausdorff Topological Space ◽  
Lacunary Sequences ◽  
Summability Methods ◽  
Borel Field ◽  
New Type

Most of the summability methods cannot be defined in an arbitrary Hausdorff topological space unless one introduces a linear or a group structure. In the present paper, using distribution functions over the Borel ?-field of the topology and lacunary sequences we define a new type of convergencemethod in an arbitrary Hausdorff topological space and we study some inclusion theorems with respect to the resulting summability method. We also investigate the inclusion relation between lacunary sequence and lacunary refinement of it.

SMALL-BOUND ISOMORPHISMS OF FUNCTION SPACES

2020 ◽  
pp. 1-18
Author(s):  
JAKUB RONDOŠ ◽  
JIŘÍ SPURNÝ
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Function Spaces ◽  
Locally Compact ◽  
Hausdorff Topological Space ◽  
Choquet Boundary ◽  
Peak Point ◽  
Peak Points ◽  
Weak Peak

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ . For $i=1,2$ , let $K_{i}$ be a locally compact (Hausdorff) topological space and let ${\mathcal{H}}_{i}$ be a closed subspace of ${\mathcal{C}}_{0}(K_{i},\mathbb{F})$ such that each point of the Choquet boundary $\operatorname{Ch}_{{\mathcal{H}}_{i}}K_{i}$ of ${\mathcal{H}}_{i}$ is a weak peak point. We show that if there exists an isomorphism $T:{\mathcal{H}}_{1}\rightarrow {\mathcal{H}}_{2}$ with $\left\Vert T\right\Vert \cdot \left\Vert T^{-1}\right\Vert <2$ , then $\operatorname{Ch}_{{\mathcal{H}}_{1}}K_{1}$ is homeomorphic to $\operatorname{Ch}_{{\mathcal{H}}_{2}}K_{2}$ . We then provide a one-sided version of this result. Finally we prove that under the assumption on weak peak points the Choquet boundaries have the same cardinality provided ${\mathcal{H}}_{1}$ is isomorphic to ${\mathcal{H}}_{2}$ .

scholarly journals The Reverse of the Intermediate Value Theorem in Some Topological Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Oussama Kabbouch ◽  
Mustapha Najmeddine
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Topological Spaces ◽  
Closed Graph ◽  
Intermediate Value Theorem ◽  
Hausdorff Topological Space ◽  
Intermediate Value

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

On topological quotient maps preserved by pullbacks or products

1970 ◽  
Vol 67 (3) ◽  
pp. 553-558 ◽  
Author(s):  
B. J. Day ◽  
G. M. Kelly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Quotient Maps ◽  
Continuous Maps

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

scholarly journals Separation Axioms in Intuitionistic Fuzzy Topological Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Amit Kumar Singh ◽  
Rekha Srivastava
Keyword(s):  
Topological Space ◽  
Left Adjoint ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

In this paper we have studied separation axiomsTi,i=0,1,2in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functorsℬ:IF-Top→BF-Topand𝒟:BF-Top→IF-Topand observe that𝒟is left adjoint toℬ.

Uniformity on generalized topological spaces

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Dipankar Dey ◽  
Dhananjay Mandal ◽  
Manabendra Nath Mukherjee
Keyword(s):  
Topological Space ◽  
Present Article ◽  
Design Methodology ◽  
Original Work ◽  
Generalized Topology ◽  
Topological Spaces ◽  
Intermediate Structure ◽  
Content Type ◽  
Completely Regular ◽  
Generalized Topological Space

PurposeThe present article deals with the initiation and study of a uniformity like notion, captioned μ-uniformity, in the context of a generalized topological space.Design/methodology/approachThe existence of uniformity for a completely regular topological space is well-known, and the interrelation of this structure with a proximity is also well-studied. Using this idea, a structure on generalized topological space has been developed, to establish the same type of compatibility in the corresponding frameworks.FindingsIt is proved, among other things, that a μ-uniformity on a non-empty set X always induces a generalized topology on X, which is μ-completely regular too. In the last theorem of the paper, the authors develop a relation between μ-proximity and μ-uniformity by showing that every μ-uniformity generates a μ-proximity, both giving the same generalized topology on the underlying set.Originality/valueIt is an original work influenced by the previous works that have been done on generalized topological spaces. A kind of generalization has been done in this article, that has produced an intermediate structure to the already known generalized topological spaces.

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