Support-Based Fuzzy Density via Support Topologies and Mapping Properties

We study a support-based notion of fuzzy density together with fuzzy weight in Lowen fuzzy topological spaces. Two equivalent descriptions of the density invariant are considered: one based on supports of fuzzy sets meeting every nonzero fuzzy open set and the other on ordinary subsets meeting every nonzero fuzzy open set in a positive degree. The invariant is characterized by the ordinary density of the associated support topology. This representation is used to distinguish results transferred from ordinary topology from phenomena that retain membership-level information. We show that the supports of a fuzzy base form an ordinary base for the support topology, yielding a structural comparison between ordinary weight and fuzzy weight. For surjective fuzzy continuous and fuzzy open mappings, the fiber-cardinality estimate is shown to be sharp, while a counterexample demonstrates that fuzzy openness cannot in general be omitted from the reverse estimate. We also construct two Lowen fuzzy topologies with the same support topology and the same support-based fuzzy density but different fuzzy weights, with genuinely non-crisp membership functions playing an essential role in one of them. Applications to finite products, group quotients and fuzzy symmetric powers are obtained, and the density and weight of lower-semicontinuous Lowen fuzzy topologies are computed.

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