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Covering dimension of fuzzy spaces. (English) Zbl 0557.54006

Covering dimension of fuzzy topological spaces is defined using the notion of \(\alpha\)-shading introduced by T. E. Gantner, R. C. Steinlage and R. H. Warren [J. Math. Anal. Appl. 62, 547-562 (1978; Zbl 0372.54001)]. Properties of that covering dimension, analogous to those of covering dimension of topological spaces, are verified. If the fuzzy space is topologically generated, then this fuzzy covering dimension agrees with the covering dimension of the generating space. Let (X,T) be a fuzzy topological space and \(0\leq \alpha <1\). A collection \(U\subset T\) is called an \(\alpha\)-shading of X if for each \(x\in X\) there exists \(u\in U\) so that \(u(x)>\alpha\). If U and V are two \(\alpha\)- shadings of X, then U is an \(\alpha\)-refinement of V if for any \(u\in U\) there exists a \(v\in V\) so that \(u\leq v\). The basic definition is: \(F\)- dim\({}_{\alpha}X\leq n\) if every finite \(\alpha\)-shading of X has an \(\alpha\)-refinement of order \(\leq n\).
Reviewer: B.L.Madison

MSC:

54A40 Fuzzy topology
54F45 Dimension theory in general topology

Citations:

Zbl 0372.54001
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