Soft Bitopological Spaces via Soft Elements

TL;DR

This paper introduces soft bitopological spaces based on soft elements, establishing separation axioms and compactness concepts, and explores their properties and relationships with classical topologies, including finiteness phenomena.

Contribution

It defines soft bitopological spaces via soft elements, introduces pairwise soft separation axioms and compactness, and analyzes their properties and equivalences with parameterwise topologies.

Findings

Pairwise soft $T_i$ axioms are equivalent to parameterwise $T_i$ axioms in canonical cases.

Finiteness of the parameter set influences the relationship between componentwise and soft compactness.

Examples demonstrate differences between induced bitopologies on soft elements and original soft bitopologies.

Abstract

We introduce soft bitopological spaces from the standpoint of soft elements. A soft bitopological space is a soft set equipped with two soft topologies. Following the classical construction of Goldar--Ray, each soft topology on $F$ induces an ordinary topology on the set $\SE(F)$ of soft elements; hence every soft bitopological space canonically determines a genuine bitopological space on $\SE(F)$. Within this setting we define pairwise soft separation axioms ($T_0$, $T_1$, $T_2$) and a notion of pairwise soft compactness, and we compare them with their parameterwise counterparts. For canonical (sectionwise generated) soft bitopologies, we show that the pairwise soft $T_i$ axioms are equivalent to the corresponding pairwise $T_i$ axioms on each parameter space. Compactness exhibits a finiteness phenomenon: when the parameter set is finite, componentwise pairwise compactness forces pairwise soft compactness, while an infinite-parameter example shows that the finiteness assumption is essential. Examples are included to clarify how the induced bitopology on $\SE(F)$ may behave differently from the original soft bitopology.