On fuzzy function spaces (Q2731741)

Four types of (Chang) fuzzy topologies on the set \(FC(Y,Z)\) of all continuous mappings from a fuzzy topological space \(Y\) into a fuzzy topological space \(Z\) are considered. Namely, they are fuzzy topologies of joint continuity type, fuzzy topologies of splitting type, point-open fuzzy topologies and compact-open type fuzzy topologies. Fuzzy topologies of the last two types are similar to fuzzy topologies introduced earlier by \textit{Y. W. Peng} [Topological structure of a fuzzy function space -- the pointwise convergent topology and compact open topology, Kexue Tongbao, Foreign Lang. Ed. 29, 289-292 (1984)] and \textit{G. Jäger} [J. Fuzzy Math. 6, No. 4, 929-939 (1998; Zbl 0922.54011)]; definitions of fuzzy topologies of the first two types involve a fixed class \({\mathcal A}\) of fuzzy topological spaces. For example:NEWLINENEWLINENEWLINEA fuzzy topology \(\tau\) on \(FC(Y,Z)\) is called \({\mathcal A}\)-jointly continuous if for every \(X\in{\mathcal A}\) continuity of a mapping \(G:X\to FC_\tau (Y,Z)\) implies the continuity of the mapping \(\widetilde G:X\times Y\to Z\) where \(\widetilde G(x,y) =G(x)(y)\);NEWLINENEWLINENEWLINEA fuzzy topology \(\tau\) on \(FC(Y,Z)\) is called \({\mathcal A}\)-splitting if continuity of a mapping \(F:X\times Y\to Z\) implies continuity of the corresponding mapping \(\widehat F:X\to FC_\tau(Y,Z)\).