Methods of generating soft topologies and soft separation axioms

TL;DR

This paper explores methods to generate soft topologies from systems of crisp topologies and examines how separation axioms are preserved or transferred between these structures, providing new insights and correcting previous results.

Contribution

It introduces new methods for generating soft topologies from crisp topologies and analyzes the transfer of separation axioms between them, extending and correcting existing literature.

Findings

One formula for generating soft topologies is more constructible.

Separation axioms can be preserved under certain conditions.

The study extends and disproves some previous results.

Abstract

The paper develops a novel analysis of mutual interactions between topology and soft topology. It is known that each soft topology produces a system of crisp (parameterized) topologies. The other way round is also possible. Namely, one can generate a soft topology from a system of crisp topologies. Different methods of producing soft topologies are discussed by implementing two formulas. Then, the relationships between the resulting soft topologies are obtained. With the help of an example, it is demonstrated that one formula is more constructible than the other. Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred to the set of crisp topologies or the opposite. To address this question, we consider the standard separation axioms and show how well these axioms can be preserved when moving from a system of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings extend and disprove some results from the literature.