Soft Bitopological Groups via Soft Elements

Abstract

Working in the soft-element (classical) viewpoint, we introduce \emph{soft bitopological groups}: soft groups endowed with two soft topologies such that the induced topologies on the set of soft elements make the soft-element group into a (classical) bitopological group. This approach unifies and simplifies continuity proofs, because the group operations become coordinatewise and standard topological-group methods apply. We organize the theory in a standard ``definitions--characterizations--properties--examples'' format. In particular, we (i) record the induced topology and induced bitopology on soft elements of a soft set; (ii) characterize soft bitopological groups by continuity of the map $(a,b)\mapsto a\ast b^{-1}$ in each induced topology; (iii) show that translations and inversion are homeomorphisms in each induced topology; (iv) collect pairwise soft separation axioms and pairwise soft compactness results (including the finiteness principle for compactness when the parameter set is finite); and (v) define soft bitopological group homomorphisms and basic invariants. Several examples illustrat that the two topologies can be independent (non-comparable) even in Hausdorff situations.