Denseness of soft spaces

Amlak I. Alajlan; Amlak I. Alajlan

doi:10.3934/math.2025993

AIMS Mathematics

2025, Volume 10, Issue 9: 22294-22313. doi: 10.3934/math.2025993

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Denseness of soft spaces

Amlak I. Alajlan ^,

Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

Received: 28 July 2025 Revised: 16 September 2025 Accepted: 22 September 2025 Published: 26 September 2025
MSC : 54E99, 54F65

In a soft environment, various structures associated with sets and topological spaces have been rigorously analyzed, including soft open (closed) sets, soft separation axioms, soft connectedness, and so on. The practical applications of soft set theory underscore its significant impact on real-world problems, offering innovative solutions that enhance decision-making processes across diverse domains. In this study, we built upon existing research by introducing new concepts within the realm of soft spaces, which served as an extension of classical topology. Our focus was on soft isolated sets and soft dense sets (spaces) in themselves. We investigated the fundamental properties and characterizations of these concepts, bolstered by rigorous proofs of theorems and relevant counterexamples. We explored the interrelations between these concepts and the notion of soft closure. Additionally, we showed that a soft topology was classified as soft dense in itself if, and only if, each of its parametric topologies was similarly classified. We also examined the behavior of soft dense spaces in themselves under various operations, including the formation of soft subspaces, the soft topological sum, and the image and inverse image under specific soft mappings.
- soft set,
- soft point,
- soft topology,
- parametric topology,
- soft limit point,
- soft isolated point,
- soft dense set in itself,
- soft dense space in itself
Citation: Amlak I. Alajlan. Denseness of soft spaces[J]. AIMS Mathematics, 2025, 10(9): 22294-22313. doi: 10.3934/math.2025993

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Abstract

In a soft environment, various structures associated with sets and topological spaces have been rigorously analyzed, including soft open (closed) sets, soft separation axioms, soft connectedness, and so on. The practical applications of soft set theory underscore its significant impact on real-world problems, offering innovative solutions that enhance decision-making processes across diverse domains. In this study, we built upon existing research by introducing new concepts within the realm of soft spaces, which served as an extension of classical topology. Our focus was on soft isolated sets and soft dense sets (spaces) in themselves. We investigated the fundamental properties and characterizations of these concepts, bolstered by rigorous proofs of theorems and relevant counterexamples. We explored the interrelations between these concepts and the notion of soft closure. Additionally, we showed that a soft topology was classified as soft dense in itself if, and only if, each of its parametric topologies was similarly classified. We also examined the behavior of soft dense spaces in themselves under various operations, including the formation of soft subspaces, the soft topological sum, and the image and inverse image under specific soft mappings.

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