Stratifications of smooth fuzzy topological spaces (Q2748641)

An \([0,1]\)-fuzzy, or smooth, topological space \((X,{\mathcal T})\) [\textit{A. P. Šostak}, Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 89-103 (1985; Zbl 0638.54007)], is called stratified, if \({\mathcal T}(\alpha_X)=1\) for each constant map \(\alpha_X: X\to [0,1]\) [\textit{Pu Paoming} and \textit{Liu Yingming}, J. Math. Anal. Appl. 76, 571-599 (1980; Zbl 0447.54006)]. Given an \([0,1]\)-fuzzy topological space \((X,{\mathcal T})\), by its stratification the author calls the pair \((X,{\mathcal T}_s)\) where \({\mathcal T}_s:[0,1]^X \to[0,1]\) is the smallest \((\leq)\) stratified \([0,1]\)-fuzzy topology dominating \({\mathcal T}\). The paper contains some results about stratification of \([0,1]\)-fuzzy topological spaces. In particular, an intrinsic characterization of stratification is obtained and it is shown that stratification of the product of \([0,1]\)-fuzzy topological spaces is equivalent to the stratification of any one of the factors of this product.