Compactness and connectedness as absolute properties in fuzzy topological spaces (Q1290595)

A topological property \(P\) is called absolute if for any \(Z \subset Y\subset X\) the space \(Z\) has the property \(P\) as the subspace of \(X\) if and only if \(Z\) has the property \(P\) as the subspace of \(Y\). It is well known that properties of compactness and connectedness in ordinary (i.e. crisp) topology are absolute. In this paper the author studies a similar problem for \([0,1]\)-topological spaces (= fuzzy topological spaces in the sense of \textit{C. L. Chang} [J. Math. Anal. Appl. 24, 182-190 (1968; Zbl 0167.51001)]; besides in the role of \(Y\) and \(Z\) can be also fuzzy subsets of a \([0,1]\)-topological space, endowed with a fuzzy topological structure induced from \(X\). The principal results of the paper are: Properties of compactness in the sense of \textit{J. J. Chadwick} [ibid. 162, No. 1, 92-110 (1991; Zbl 0772.54005)]; \(N\)-compactness in the sense of \textit{Wang Guojun} [ibid. 94, 1-23 (1983; Zbl 0512.54006)]; and connectedness in the sense of \textit{Pu Paoming} and \textit{Liu Yingming} [ibid. 76, 571-599 (1980; Zbl 0447.54006)] of fuzzy subsets of \([0,1]\)-topological spaces are absolute.