Symmetric power functors in the category of fuzzy compact spaces

In this paper, we establish that inverse limits of fuzzy compact spaces remain fuzzy compact, using a direct proof based solely on Lowen’s definitions. This result enables a categorical treatment of compactness analogous to the Tychonoff theorem in the classical setting. Moreover, we prove that the symmetric power functor is normal in the sense adapted to the category of fuzzy compact spaces. It preserves inverse limits of surjective systems, weight, intersections and preimages. It respects embeddings and surjections and it behaves correctly on the empty and one-point spaces. Indeed, we show by these results that the fuzzy symmetric power construction faithfully generalizes its classical counterpart while preserving the essential structural and categorical properties of compactness.

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