Soft Bitopological Groups via Soft Elements
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917272490082304 |
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| author | Ray, S. |
| author_facet | Ray, S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12678 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Soft Bitopological Groups via Soft Elements Ray, S. General Topology 03E72, 54A40, 54H11, 22A30 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. |
| title | Soft Bitopological Groups via Soft Elements |
| topic | General Topology 03E72, 54A40, 54H11, 22A30 |
| url | https://arxiv.org/abs/2602.12678 |