On the impact of the soft symmetric power functor SPⁿ on soft almost continuous and soft almost open mappings
Функтор мягкой симметрической степени
DOI:
https://doi.org/10.56143/yarnhj89Abstract
In this paper, we study the behavior of soft almost continuous and soft almost open mappings under two natural constructions in soft topology: finite Cartesian powers and soft symmetric powers. We first show that both soft almost continuity and soft almost openness are preserved under finite powers equipped with the soft product topology. We then consider the induced mappings on the n-th soft symmetric power SPⁿ, viewed as the quotient of the n-fold soft product under the natural action of the symmetric group. Unlike the classical case, the canonical soft symmetric projection need not be soft open. In the symmetric-power setting, we prove that soft almost continuity is preserved by the induced mapping on SPⁿ. For soft almost openness, we obtain a sufficient condition on the domain-side canonical quotient map ensuring that the induced mapping on SPⁿ remains soft almost open. Thus, preservation is unconditional for finite powers, while in the symmetric-power setting the almost-open case is established under an explicit additional hypothesis. These results clarify the role of quotient constructions in the soft setting and further illustrate the difference between classical and soft symmetric powers.
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