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On $e^*$-$θ$-$D$-sets and Related Topics

D. Akalın, M. Özkoç

Abstract

This paper aims to study the notion of $e^*\text{-}θ$-open \cite{Ozkoc1} sets and to investigate new properties of this notion. Also, we define a new type of set, called $e^*\text{-}θ\text{-}D$-set, via the notion of $e^*\text{-}θ$-open set. Moreover, we introduce some new separation axioms by utilizing $e^*\text{-}θ\text{-}D$-sets. We obtain many results related to these new notions. In addition, the notions of $e^*\text{-}θ\text{-}$kernel and slightly $e^*\text{-}θ\text{-}R_0$ space are defined. Some characterizations regarding these new notions have been obtained. Furthermore, we have given many examples concerning the mentioned notions. Finally, we not only put forward the definition of $e^*\text{-}R_1$ space but also obtained some of its characterizations.

On $e^*$-$θ$-$D$-sets and Related Topics

Abstract

This paper aims to study the notion of -open \cite{Ozkoc1} sets and to investigate new properties of this notion. Also, we define a new type of set, called -set, via the notion of -open set. Moreover, we introduce some new separation axioms by utilizing -sets. We obtain many results related to these new notions. In addition, the notions of kernel and slightly space are defined. Some characterizations regarding these new notions have been obtained. Furthermore, we have given many examples concerning the mentioned notions. Finally, we not only put forward the definition of space but also obtained some of its characterizations.
Paper Structure (7 sections, 36 theorems, 5 equations)

This paper contains 7 sections, 36 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On $e^*\text{-}\theta$-open sets
  4. On $e^*\text{-}\theta\text{-}D_{i}$ and $e^*\theta\text{-}T_{i}$ topological spaces
  5. Further properties
  6. $e^*\text{-}R_1$ space
  7. Acknowledgements

Key Result

Theorem 2.1

$($Ozkoc1$)$ The following properties hold for a subset $A$ of a topological space $(X , \tau):$$(a)$$A\in e^*O(X)$ if and only if $e^*\text{-}cl(A)\in e^*R(X),$$(b)$$A\in e^*C(X)$ if and only if $e^*\text{-}int(A)\in e^*R(X).$

Theorems & Definitions (99)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 89 more