On β-Open Sets and Ideals in Topological Spaces

European Journal of Pure and Applied Mathematics, 2019

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 β I-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 β I-compact space if it is β I-compact as a subset. An ideal topological space (X, τ, I) is said to be a cβ I-compact space if it is cβ I-compact as a subset. An ideal topological space (X, τ, I) is said to be a countably β I-compact space if X is countably β I-compact as a subset. Two sets A and B in an ideal topological space (X, τ, I) is said to be β I-separated if cl β I (A) ∩ B = ∅ = A ∩ cl β (B). A subset A of an ideal topological space (X, τ, I) is said to be β I-connected if it cannot be expressed as a union of two β I-separated sets. An ideal topological space (X, τ, I) is said to be β I-connected if X β I-connected as a subset. In this study, we introduced the notions β I-open set, β I-compact, cβ I-compact, β I-hyperconnected, cβ I-hyperconnected, β I-connected and β I-separated. Moreover, we investigated the concept β-open set by determining some of its properties relative to the above-mentioned notions. 2010 Mathematics Subject Classifications: 54-XX Key Words and Phrases: β-open sets, β I-open sets, β I-compactness, cβ I-compactness, β Ihyperconnectedness and cβ I-hyperconnectednes * Corresponding author.

2017

In this paper, first we give some characterizations and properties of strong β-I-open sets. Then, we obtain a decomposition of semi-I-closed sets by using strong β-I-open sets and t-I-sets. In addition, we give decompositions of open sets in any extremally disconnected space and in any ideal topological space. Moreover, we define AKI -set and AK ∗ I -set and give decompositions of continuity.

In this paper, we investigated some properties of a δα − I − open set [6] and a semi * − I − open set [6] in ideal topological spaces. Moreover, the relationships of other related classes of sets are investigated. Also, a new decomposition of continuous functions is obtained by using δ − β − I−continuous and S * − continuous functions.

Bol. Soc. Paran. Mat., 2023

In this paper we introduce and investigate some properties of semi *-I-open sets, pre *-I-open sets and e-I-open sets in ideal topological spaces. Moreover, some relationships among semi *-I-open sets, e-I-open sets and pre *-I-open sets in ideal topological spaces are established. Finally, we obtain the decompositions of continuity.

2016

In this paper, the notions of δI -semi-open sets and δI -semi-closed sets are introduced and investigated in ideal topological spaces. MSC: 54A05.

AL-Rafidain Journal of Computer Sciences and Mathematics

Malaya Journal of Matematik, 2018

In this paper, we study semi-open, pre-open, α-open and β-open sets, and obtain some relations between them.

Journal of Southwest Jiaotong University

In this research work, we introduce the concept of countably α-compact spaces in an ideal topological space (X,τ,I) and study further properties of α-continuous functions. We prove that α-continuous function mapping a countably α-compact ideal topological space (X,τ,I) to the ideal space (Y,σ,J) is an α-closed subset of the Cartesian product (X×Y,τ×σ,I×J). We showed that the countably α-compact ideal space (X,τ,I) with weight ∣X∣≥ ℵ₀ is the α-continuous image of a closed subspace of the cube D^({ℵ₀}) and illustrate that the α-continuous function f:(X,τ,I)→(Y,σ,J) where Y is countably α-compact can be extended over its domain under some constraints. Moreover, α-pseudocompact is defined in an ideal topological space (X,τ,I) and we proved that countably α-pseudocompactness is not hereditary with respect to α-closed sets and we showed that countab α-pseudocompactness is not finitely multiplicative. We find that if the ideal topological space (X,τ,I) is Tychonoff, (Y,σ,J) is countably α-...

2016

In this paper, new classes of sets in general topology called a supra-β-open (closed) ( an infra-β-open (closed) ) set are introduced. Using new concepts, the fundamental properties and special results are highlighted. The relations between supra-β-open (closed) ( an infra-β-open (closed) ) set and other topological sets are investigated. Moreover, counter-examples are given to show that the converse of these relations in Diagram 1 need not be true, in general. Finally, some special theorems are introduced by adding condition to achieve the converse relations in Diagram 1.