\(H(\lambda)\)-completely regular \(L\)-fuzzy sets and their applications
Let \(I(L)\) denote the Hutton unit interval with the usual fuzzy topology, and \(\widetilde{I}(L)\) denote the fuzzy topological space with the same underlying set of \(I(L)\) and with the fuzzy topology generated as a subbase by the union of the usual fuzzy topology on \(I(L)\) and the topological modification of the usual fuzzy topology, namely the fuzzy topology on \(\widetilde{I}(L)\) is the coarsest weakly induced fuzzy topology finer than the usual fuzzy topology. Replacing \(I(L)\) by \(\widetilde{I}(L)\) in the defintion of fuzzy completely regular spaces one gets the \(H(\lambda)\) completely regular spaces. Some characterizations of \(H(\lambda)\) complete regularity are given in this paper parallel to those for complete regularity. It is essentially proved that the category of \(H(\lambda)\) completely regular spaces is concretely reflective in the category of weakly induced spaces. It must be pointed out that Definition 3.10 in the paper is not well-defined, since generally \(\overline{\delta}\) is not \(H(\lambda)\) completely regular, otherwise, let \(L=2\), we get that the category of completely regular spaces is both reflective and coreflective in the category of topological spaces, but this is impossible [\textit{V. Kannan}, Math. Ann. 195, 168-174 (1972; Zbl 0215.09701)].
scientific article; zbMATH DE number 1305157