Separation axioms via novel operators in the frame of topological spaces and applications

Mesfer H. Alqahtani; Alaa M. Abd El-latif; Mesfer H. Alqahtani; Alaa M. Abd El-latif

doi:10.3934/math.2024690

AIMS Mathematics

2024, Volume 9, Issue 6: 14213-14227. doi: 10.3934/math.2024690

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Separation axioms via novel operators in the frame of topological spaces and applications

Mesfer H. Alqahtani ¹,
Alaa M. Abd El-latif ^{2
,
,}

1.
Department of Mathematics, University College of Umluj, University of Tabuk, Tabuk 48322, Saudi Arabia
2.
Mathematics Department, College of Sciences and Arts, Northern Border University, Rafha 91911, Saudi Arabia

Received: 08 February 2024 Revised: 11 April 2024 Accepted: 12 April 2024 Published: 18 April 2024
MSC : 54A05, 54C05, 54C08

In this work, we introduce a very wide category of open sets in topological spaces, called $ \aleph $-open sets. We study the category of $ \aleph $-open sets that contains $ \beta $-open sets in addition to $ \beta^{\ast} $-open and $ e^{\ast} $-open sets. We present the essential properties of this class and disclose its relationships with many different classes of open sets with the help of concrete counterexamples. In addition, we introduce the $ \aleph $-interior and $ \aleph $-closure operators. Moreover, we study the concept of $ \aleph $-continuity of functions inspired by the classes of $ \aleph $-open and $ \aleph $-closed sets. Also, we discuss some kinds of separation axioms and some theorems related to the graph of functions.
- $ \aleph $-open set,
- $ \aleph $-continuous function,
- graph of functions,
- $ \aleph $-$ T_0 $-space,
- $ \aleph $-$ T_1 $-space,
- $ \aleph $-$ T_2 $-space,
- applications
Citation: Mesfer H. Alqahtani, Alaa M. Abd El-latif. Separation axioms via novel operators in the frame of topological spaces and applications[J]. AIMS Mathematics, 2024, 9(6): 14213-14227. doi: 10.3934/math.2024690

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Abstract

In this work, we introduce a very wide category of open sets in topological spaces, called $ \aleph $-open sets. We study the category of $ \aleph $-open sets that contains $ \beta $-open sets in addition to $ \beta^{\ast} $-open and $ e^{\ast} $-open sets. We present the essential properties of this class and disclose its relationships with many different classes of open sets with the help of concrete counterexamples. In addition, we introduce the $ \aleph $-interior and $ \aleph $-closure operators. Moreover, we study the concept of $ \aleph $-continuity of functions inspired by the classes of $ \aleph $-open and $ \aleph $-closed sets. Also, we discuss some kinds of separation axioms and some theorems related to the graph of functions.

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