Soft Bitopological Groups via Soft Elements

Abstract

Let $F$ be a soft group over a parameter set $A$. We equip the selector group $\mathrm{SE}(F)=\prod_{t\in A}F(t)$ with the topology generated by componentwise open boxes. This construction replaces a sectionwise rule that does not define a topology on arbitrary selector subsets. It depends only on the component topologies and may therefore lose correlations between parameters. A soft bitopological group is then a soft group with two selector topologies, each making $\mathrm{SE}(F)$ a topological group. We also study a soft paratopological group together with its inverse topology. In this conjugate pair, inversion interchanges the two selector topologies, and equality of the two is equivalent to continuity of inversion. We prove componentwise separation criteria, finite-parameter compactness and $\omega$-boundedness results, and infinite-parameter counterexamples caused by the box topology. We also characterize when a soft union creates mixed selectors outside the two original selector sets.