Some topological aspects of interval spaces

Muhammad Qasim; Arbaz Jehan Khan; Samirah Alsulami; Shoaib Assar; Muhammad Qasim; Arbaz Jehan Khan; Samirah Alsulami; Shoaib Assar

doi:10.3934/math.2023190

AIMS Mathematics

2023, Volume 8, Issue 2: 3826-3841. doi: 10.3934/math.2023190

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Research article

Some topological aspects of interval spaces

1.
Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), H-12 Islamabad, Pakistan
2.
Department of Mathematics, College of Science, University of Jeddah, Jeddah 21577, Saudi Arabia

Received: 04 October 2022 Revised: 15 November 2022 Accepted: 22 November 2022 Published: 29 November 2022
MSC : 54A05, 54B30, 54D10, 54E99, 54F45

In previous papers, several $ T_{0} $, $ T_{2} $ objects, $ D $-connectedness and zero-dimensionality in topological categories have been introduced and compared. In this paper, we characterize separated objects, $ T_{0} $, $ {\bf{T_{0}}} $, $ T_{1} $, Pre-$ T_{2} $ and several versions of Hausdorff objects in the category of interval spaces and interval-preserving mappings and examine their mutual relationship. Further, we give the characterization of the notion of closedness and $ D $-connectedness in interval spaces and study some of their properties. Finally, we introduce zero-dimensionality in this category and show its relation to $ D $-connectedness.
- interval space,
- convex space,
- separated, Hausdorff,
- zero-dimensional,
- initial lift,
- topological category
Citation: Muhammad Qasim, Arbaz Jehan Khan, Samirah Alsulami, Shoaib Assar. Some topological aspects of interval spaces[J]. AIMS Mathematics, 2023, 8(2): 3826-3841. doi: 10.3934/math.2023190

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Abstract

In previous papers, several $ T_{0} $, $ T_{2} $ objects, $ D $-connectedness and zero-dimensionality in topological categories have been introduced and compared. In this paper, we characterize separated objects, $ T_{0} $, $ {\bf{T_{0}}} $, $ T_{1} $, Pre-$ T_{2} $ and several versions of Hausdorff objects in the category of interval spaces and interval-preserving mappings and examine their mutual relationship. Further, we give the characterization of the notion of closedness and $ D $-connectedness in interval spaces and study some of their properties. Finally, we introduce zero-dimensionality in this category and show its relation to $ D $-connectedness.

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