Abstract
Soft topology studies a structure on the collection of all soft sets on a given set of alternatives (the relevant attributes being fixed). It is directly inspired by the axioms of a topological space. This paper contributes to the theoretical bases of soft topology in various ways. We extend a general construction of soft topologies from topologies on the set of alternatives in two different directions. An extensive discussion with criteria about what a soft counterpart of “topological separability” should satisfy is also given. The interactions of the properties that arise with separability, and of second-countability and its soft counterpart, are studied under the general mechanisms that generate soft topological spaces. The first non-trivial examples of soft second-countable soft topological spaces are produced as a consequence.
Highlights
Soft topology stands at the junction of soft set theory [1] and topology [2,3]
Soft second-countability, which is quite indisputable as an axiom in soft topology, is strictly stronger than these concepts, and we prove by example that it is different from them
We say that τ (Σ) is the soft topology on X generated by the family of topologies Σ
Summary
Soft topology stands at the junction of soft set theory [1] and topology [2,3]. It is concerned with a structure on the set of all soft sets, and is inspired by the standard axioms of a topological space. We propose some axioms of ‘soft separability’ and study their behavior and the theoretical properties of soft second-countability. They rely on completely different approaches to the idea of ‘soft separability’, they are not independent. Their relationship with topological separability is analyzed through the (improved) process that passes topologies on to soft topologies. It allows us to visualize soft topologies in a simpler manner because bases are simpler than topologies and the same is true in the field of soft topologies Another advantage is that it is a very natural way for the analysis of soft second-countability. The last section summarizes this study and suggests possible future works
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