On the topology $ \tau^{\diamond}_R $ of primal topological spaces

Murad ÖZKOÇ; Büşra KÖSTEL; Murad ÖZKOÇ; Büşra KÖSTEL

doi:10.3934/math.2024834

AIMS Mathematics

2024, Volume 9, Issue 7: 17171-17183. doi: 10.3934/math.2024834

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Research article

On the topology $ \tau^{\diamond}_R $ of primal topological spaces

Murad ÖZKOÇ ^{1
,
,},
Büşra KÖSTEL ²

1.
Muğla Sıtkı Koçman University, Faculty of Science, Department of Mathematics, 48000, Menteşe-Muğla, Turkey
2.
Muğla Sıtkı Koçman University, Graduate School of Natural and Applied Sciences, Mathematics, 48000, Menteşe-Muğla, Turkey

Received: 25 February 2024 Revised: 19 April 2024 Accepted: 26 April 2024 Published: 17 May 2024
MSC : 54A05, 54B99, 54C60

The main purpose of this paper is to introduce and study two new operators $ (\cdot)_R^{\diamond} $ and $ cl_R^{\diamond}(\cdot) $ via primal, which is a new notion. We show that the operator $ cl_R^{\diamond}(\cdot) $ is a Kuratowski closure operator, while the operator $ (\cdot)_R^{\diamond} $ is not. In addition, we prove that the topology on $ X $, shown as $ \tau_R^{\diamond}, $ obtained by means of the operator $ cl_R^{\diamond}(\cdot), $ is finer than $ \tau_{\delta}, $ where $ \tau_{\delta} $ is the family of $ \delta $-open subsets of a space $ (X, \tau). $ Moreover, we not only obtain a base for the topology $ \tau_R^{\diamond} $ but also prove many fundamental results concerning this new structure. Furthermore, we provide many counterexamples related to our results.
- primal,
- primal topological space,
- Kuratowski closure operator,
- the operator $ (\cdot)_R^{\diamond} $,
- the operator $ cl^{\diamond}_R $
Citation: Murad ÖZKOÇ, Büşra KÖSTEL. On the topology $ \tau^{\diamond}_R $ of primal topological spaces[J]. AIMS Mathematics, 2024, 9(7): 17171-17183. doi: 10.3934/math.2024834

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Abstract

The main purpose of this paper is to introduce and study two new operators $ (\cdot)_R^{\diamond} $ and $ cl_R^{\diamond}(\cdot) $ via primal, which is a new notion. We show that the operator $ cl_R^{\diamond}(\cdot) $ is a Kuratowski closure operator, while the operator $ (\cdot)_R^{\diamond} $ is not. In addition, we prove that the topology on $ X $, shown as $ \tau_R^{\diamond}, $ obtained by means of the operator $ cl_R^{\diamond}(\cdot), $ is finer than $ \tau_{\delta}, $ where $ \tau_{\delta} $ is the family of $ \delta $-open subsets of a space $ (X, \tau). $ Moreover, we not only obtain a base for the topology $ \tau_R^{\diamond} $ but also prove many fundamental results concerning this new structure. Furthermore, we provide many counterexamples related to our results.

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