Soft Bitopological Spaces via Soft Elements
S. Ray
TL;DR
This work develops soft bitopological spaces by equipping a soft set $F$ with two soft topologies and studying the induced classical bitopology on the set of soft elements $\mathrm{SE}(F)$ via the topologies $(\tau_1)^*$ and $(\tau_2)^*$. It defines pairwise soft separation axioms ($T_0$, $T_1$, $T_2$) and pairwise soft compactness, and shows that in canonical (sectionwise generated) soft bitopologies these properties correspond exactly to their parameterwise counterparts on each fiber $F(t)$. A key finiteness phenomenon is established: when the parameter set $A$ is finite, componentwise pairwise compactness implies pairwise soft compactness, while this implication can fail for infinite $A$. The paper also demonstrates that separation properties of the induced topology on $\mathrm{SE}(F)$ can differ from those of the original soft bitopology and provides a reconstruction framework to derive canonical soft structures from bitopological data on $\mathrm{SE}(F)$. Overall, it bridges soft-set theory with classical bitopology, enabling transfer of methods between the two settings and guiding future work on further pairwise properties and product constructions.
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.