scholarly journals Generalized topologies with associating function and logical applications

2020 ◽  
Vol 24 (2) ◽  
pp. 253-269
Author(s):  
Tomasz Witczak
Keyword(s):  
Modal Logic ◽  
Topological Space ◽  
Special Function ◽  
Open Set ◽  
Generalized Topological Space

The whole universe of a generalized topological space may not be open. Hence, some points may be beyond any open set. In this paper we assume that such points are associated with certain open neighbourhoods by means of a special function F. We study various properties of the structures obtained in this way. We introduce the notions of F-interior and F-closure and we discuss issues of convergence in this new setting. It is possible to treat our spaces as a semantical framework for modal logic.

scholarly journals A covering property with respect to generalized preopen sets

2019 ◽  
Vol 38 (6) ◽  
pp. 25-32
Author(s):  
Ajoy Mukharjee
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Covering Property ◽  
Open Set ◽  
Generalized Topological Space ◽  
Preopen Set

In this paper,  we introduce and study the notion of $\mu$-precompact spaces on the observation that  each $\mu$-preopen set of a generalized topological space is contained  in a $\mu$-open set. The $\mu$-precompactness is weaker than $\mu$-compactness but stronger than weakly $\mu$-compactness of  generalized topological spaces.

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.

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 On generalized γµ-closed sets and related continuity

2021 ◽  
Vol 13 (2) ◽  
pp. 483-493
Author(s):  
Ritu Sen
Keyword(s):  
Topological Space ◽  
Main Interest ◽  
Closed Sets ◽  
Separation Axioms ◽  
Preservation Theorems ◽  
Generalized Topological Space ◽  
New Type ◽  
Weak Separation

Abstract In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

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.

Δμ-sets and ∇μ-sets in generalized topological spaces

2017 ◽  
Vol 24 (3) ◽  
pp. 403-407
Author(s):  
Pon Jeyanthi ◽  
Periadurai Nalayini ◽  
Takashi Noiri
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Sets ◽  
Generalized Topological Space

AbstractIn this paper, we introduce and study some properties of the sets, namely {\Delta_{\mu}}-sets, {\nabla_{\mu}}-sets and {\Delta_{\mu}^{*}}-closed sets in a generalized topological space.

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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