scholarly journals Pair-Countable, Closure-Preserving Covers of Compact Sets

1988 ◽  
Vol 44 (1) ◽  
pp. 138-142
Author(s):  
Henry Potoczny
Keyword(s):  
Topological Space ◽  
Compact Sets ◽  
Open Set ◽  
Open Subspace

AbstractIn this paper, we prove the following results: (1) if a topological space X has a pair-countable, closure-preserving cover of compact sets, then X is locally paracompact at each point of X and X has a dense open subspace which is locally σ-compact. In addition, if X is also collectionwise-T2, then X is paracompact. Locally paracompact is taken to mean that each point X has an open set with paracompact closure.

scholarly journals Some Properties of Interior and Closure in General Topology

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 624
Author(s):  
Soon-Mo Jung ◽  
Doyun Nam
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Open Set ◽  
Necessary And Sufficient ◽  
Open Subspace

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

A Note on Certain Spaces with Bases (mod K)

1975 ◽  
Vol 27 (2) ◽  
pp. 469-474
Author(s):  
Harold R. Bennett ◽  
Harold W. Martin
Keyword(s):  
Topological Space ◽  
Open Covering ◽  
Compact Sets ◽  
Natural Numbers ◽  
Open Set

In this note all spaces are assumed to be regular T1 spaces and all undefined terms and notations may be found in [8], In particular let cl(A) denote the closure of the set A and let Z+ denote the set of natural numbers.Definition 1. Let X be a topological space and a covering of X by compact sets. An open covering of X is said to be a basis (mod K) if whenever and an open set V contains Kx, then there exists such that . In such a case X is written as the ordered triple .

scholarly journals On s-g-cocompact open set and Continuity

2020 ◽  
Vol 25 (2) ◽  
pp. 67-77 ◽  
Author(s):  
Raad Al-Abdulla ◽  
Salam Jabar
Keyword(s):  
Topological Space ◽  
Open Set ◽  
The Relationship

    Throughout this paper by a space we mean a supra topological space, we have studied some of propertiese to new set is called supra generalize- cocompact open set ( -g-coc-open set)and find the relation with other sets and our concluded anew class of the function called -g-coc-continuous, -g-coc'-continuous, -coc-continuous, -coc'-continuous We shall provided some properties of these concepts and it will explain the relationship among them and some results on this subjects are proved Throughout this work , and new concept have been illustrated including , -coc-ompact space .

scholarly journals On B-Open Sets

2019 ◽  
Vol 12 (2) ◽  
pp. 358-369
Author(s):  
Layth Muhsin Habeeb Alabdulsada
Keyword(s):  
Topological Space ◽  
Open Set

The aim of this paper is to introduce and study $\mathcal{B}$-open sets and related properties. Also, we define a bi-operator topological space $(X, \tau, T_1, T_2)$, involving the two operators $T_1$ and $T_2$, which are used to define $\mathcal{B}$-open sets. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is, in turn, a generalization of a pre-open set and a semi-open set. We introduce a number of concepts based on $\mathcal{B}$-open sets.

scholarly journals Paracompactness with respect to an ideal

1997 ◽  
Vol 20 (3) ◽  
pp. 433-442 ◽  
Author(s):  
T. R. Hamlett ◽  
David Rose ◽  
Dragan Janković
Keyword(s):  
Topological Space ◽  
Finite Union ◽  
Open Cover ◽  
Locally Finite ◽  
Open Set ◽  
Special Cases ◽  
Paracompact Spaces

An ideal on a setXis a nonempty collection of subsets ofXclosed under the operations of subset and finite union. Given a topological spaceXand an idealℐof subsets ofX,Xis defined to beℐ-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all ofXexcept for a set inℐ. Basic results are investigated, particularly with regard to theℐ- paracompactness of two associated topologies generated by sets of the formU−IwhereUis open andI∈ℐand⋃{U|Uis open andU−A∈ℐ, for some open setA}. Preservation ofℐ-paracompactness by functions, subsets, and products is investigated. Important special cases ofℐ-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of Singal and Arya [“On m-paracompact spaces”, Math. Ann., 181 (1969), 119-133].

scholarly journals Lebesgue Measurability of Separately Continuous Functions and Separability

2007 ◽  
Vol 2007 ◽  
pp. 1-4
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Continuous Functions ◽  
Negative Answer ◽  
Regular Space ◽  
Property A ◽  
Completely Regular ◽  
Open Set ◽  
Countable Chain Condition ◽  
Countable Chain

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

A View on Fuzzy Minimal Open Sets and Fuzzy Maximal Open Sets

2015 ◽  
Vol 7 (1) ◽  
pp. 62-73 ◽  
Author(s):  
Hamid Reza Moradi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Basic Properties ◽  
New Class ◽  
Properties And Characterization ◽  
Open Set ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces ◽  
Characterization Theorems

A nonzero fuzzy open set () of a fuzzy topological space is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in is either or itself (resp. either or itself). In this note, a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied which are subclasses of open sets. Some basic properties and characterization theorems are also to be investigated.

N-BODY HAMILTONIANS WITH HARD-CORE INTERACTIONS

1994 ◽  
Vol 06 (04) ◽  
pp. 515-596 ◽  
Author(s):  
ANNE BOUTET de MONVEL-BERTHIER ◽  
VLADIMIR GEORGESCU ◽  
AVY SOFFER
Keyword(s):  
Maximal Regularity ◽  
Hard Core ◽  
Regularity Result ◽  
Weak Sense ◽  
Compact Sets ◽  
Dirichlet Laplacian ◽  
Irregular Boundary ◽  
Open Set ◽  
Dilation Group ◽  
Threshold Energies

We study N-body hamiltonians with short and long range potentials which are infinite on compact sets of non-zero measure. We show that the generator of the dilation group is locally conjugated to them away from the threshold energies. The notion of conjugacy has to be interpreted in a very weak sense, but this is enough to deduce an optimal form of the limiting absorption principle, and so absence of singular continuous spectrum and local decay. One of the main technical steps of our approach requires a maximal regularity result for the Dirichlet Laplacian in an open set with irregular boundary. We prove it for a large class of non-smooth domains.

scholarly journals On âg Compact Spaces

2014 ◽  
Vol 10 ◽  
pp. 19-22
Author(s):  
V. Senthilkumaran ◽  
R. Krishnakumar ◽  
Y. Palaniappan
Keyword(s):  
Topological Space ◽  
Compact Sets ◽  
Compact Spaces

We introduce and study some subsets of a topological space X called âg compact sets; âg compact spaces are defined and their properties are studied.

scholarly journals On β-Open Sets and Ideals in Topological Spaces

2019 ◽  
Vol 12 (3) ◽  
pp. 893-905
Author(s):  
Glaisa T. Catalan ◽  
Roberto N. Padua ◽  
Michael Jr. Patula Baldado
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Topological Spaces ◽  
Open Set ◽  
Separated Sets ◽  
The Ideal

Let X be a topological space and I be an ideal in X. A subset A of a topological space X is called a β-open set if A ⊆ cl(int(cl(A))). A subset A of X is called β-open with respect to the ideal I, or βI -open, if there exists an open set U such that (1) U − A ∈ I, and (2) A − cl(int(cl(U))) ∈ I. A space X is said to be a βI -compact space if it is βI -compact as a subset. An ideal topological space (X, τ, I) is said to be a cβI -compact space if it is cβI -compact as a subset. An ideal topological space (X, τ, I) is said to be a countably βI -compact space if X is countably βI -compact as a subset. Two sets A and B in an ideal topological space (X, τ, I) is said to be βI -separated if clβI (A) ∩ B = ∅ = A ∩ clβ(B). A subset A of an ideal topological space (X, τ, I) is said to be βI -connected if it cannot be expressed as a union of two βI -separated sets. An ideal topological space (X, τ, I) is said to be βI -connected if X βI -connected as a subset. In this study, we introduced the notions βI -open set, βI -compact, cβI -compact, βI -hyperconnected, cβI -hyperconnected, βI -connected and βI -separated. Moreover, we investigated the concept β-open set by determining some of its properties relative to the above-mentioned notions.

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