On a topological model for the category of fuzzy topological spaces. I (Q915172)

For a given map \(U:X\to [0,1]\), let \(H(U)=\{(x,t)\in X\times [0,1):\) \(t<U(x)\}\). After \textit{R. Lowen} [J. Math. Anal. Appl. 64, 446-454 (1978; Zbl 0381.54004); cf. also \textit{S. E. Rodabaugh}, Topology Appl. 11, 319- 334 (1980; Zbl 0484.54008)]) one knows that the category of fuzzy topological spaces embeds into TOP via the so called hypergraph functor: (X,\(\tau\))\(\to (X\times [0,1),\{H(U):U\in \tau \})\) and (f: \(X\to Y)\to (f\times id:\) \(X\times [0,1)\to Y\times [0,1))\). A number of topological invariants are lattice-theoretic properties of the family of all open sets. Definitions of such concepts automatically translate into the family of all open fuzzy subsets. The author makes the observation that properties of this type are preserved by the hypergraph functor. \{Reviewer's remark: One should note that the case of compactness has already been noticed by R. Lowen [op.cit.]. Also, Propositions 1.7, 1.8 and 3.1 are obvious (and well-known) in fuzzy-TOP and need not be proved in TOP using the hypergraph functor.\}