Pairwise Separation and Compactness in Soft Bitopological Spaces via Soft Elements

Abstract

Let $F:A\to \mathcal{P}(X)$ be a soft set, and let $SE(F)$ be the set of all maps $a:A\to X$ such that $a(t)\in F(t)$, for every $t\in A$. If $\tau$ is a soft topology on $F$, the sets $SE(H)$, where $H\in \tau$, form a basis for an ordinary topology ${\tau }^{\mathrm{e}}$ on $SE(F)$, called the \emph{induced soft-element topology}. Hence, two soft topologies on $F$ induce a bitopological space on $SE(F)$. We define pairwise soft $T_0$, $T_1$, and $T_2$ axioms using soft elements and prove that they are equivalent to the corresponding pairwise axioms of the induced soft-element bitopological space. For canonical soft bitopologies, these properties are also equivalent to the same axioms in every component bitopological space. We also study pairwise soft compactness. For canonical soft bitopological spaces, pairwise soft compactness implies pairwise compactness of every component space, and the converse holds when the parameter set is finite. An example with infinitely many parameters shows that the finiteness condition is needed.

Revised version: references corrected and updated; assumptions and contraction conditions clarified