On the Impact of the Soft Symmetric Power Functor SP n on Soft Almost-Open and Soft Pseudo-Open Mappings
DOI:
https://doi.org/10.58649/k9c0cj74Keywords:
soft set, soft topology, soft almost continuity, soft almost openness, product soft topology, symmetric powerAbstract
In this paper, we investigate the behavior of the soft symmetric power functor SPn with respect to two classes of soft mappings, namely soft almost-open mappings and soft pseudo-open mappings. After recalling the construction of the soft product topology and the n-th soft symmetric power, we show that the canonical soft quotient projection onto a soft symmetric power need not be soft open, which reveals an essential difference from the classical setting. Under the additional assumption that the canonical soft projection on the codomain side is soft open, we prove that the induced mapping (SPnf)SPnψ preserves soft almost-openness. We also establish a preservation theorem for soft pseudo-open mappings under SPn provided that each fiber fψ-1(yk) is soft compact and the canonical soft projection on the codomain side is soft open. Finally, we formulate an open problem concerning the necessity of the fiberwise soft compactness assumption in the pseudo-open case.
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