Soft Bitopological Spaces via Soft Elements

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.