Level-Density Functions and Cardinal Spectra in Lowen Fuzzy Topological Spaces

Classical level constructions extract an ordinary topology from a fuzzy topology at one threshold, but a single level does not measure how the cardinal complexity of dense sets changes as the threshold varies. Motivated by this loss of inter-level information, we associate with every Lowen fuzzy topological space (X,S) the level-density function δ(X,S)(a)=ω+dX,ιa(S), a∈[0,1), together with its range and supremum. This threshold-sensitive profile distinguishes fuzzy structures having the same zero-level topology. We prove that strict-level formation commutes exactly with finite fuzzy products and, under a natural openness hypothesis, with quotient formation; orbit quotients automatically satisfy this hypothesis. We also show that the profile need not be monotone, that arbitrary finite and countable sequences of infinite cardinals can be realized on prescribed half-open partitions of [0,1), and that the supremum of the spectrum need not be attained. All realization results are proved in ZFC and require neither CH nor GCH. As an application of the product and quotient identities, finite fuzzy symmetric powers preserve the entire level-density function. Thus the invariant exhibits substantial inter-level flexibility while remaining rigid under several natural fuzzy-topological constructions.