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Metrization theorems in \(L\)-topological spaces. (English) Zbl 1070.54007
With \(({\mathbf L},\vee,\wedge^I)\) a completely distributive lattice with an order-reversing involution, \({\mathbf X}\) a nonempty set, \({\mathbf L}^{{\mathbf X}}\) the set of all \({\mathbf L}\)-fuzzy sets on \({\mathbf X}\), \({\mathbf M}({\mathbf L}^{{\mathbf X}})\) the set of all nonzero \(\vee\)- irreducible elements in \({\mathbf L}^{{\mathbf X}}\), \(\underline 0\) and \(\underline 1\) the smallest element and largest element in \({\mathbf L}^{{\mathbf X}}\), the authors start with an \({\mathbf L}\)-topological space \(({\mathbf X},\eta)\) in which \(\eta\) is an \({\mathbf L}\)-co-topology in \({\mathbf X}\); an element of \(\eta\), called a closed \({\mathbf L}\)-set, is also called a \({\mathbf R}\)-neighbourhood of \({\mathbf e}= {\mathbf M}({\mathbf L}^{{\mathbf X}})\) if \({\mathbf e}\leq{\mathbf P}\); a map \({\mathbf f}: {\mathbf M}({\mathbf L}^{{\mathbf X}})\to{\mathbf L}^{{\mathbf X}}\) is called an \({\mathbf R}\)-map on \({\mathbf L}^{{\mathbf X}}\), if for all \({\mathbf a}\in{\mathbf M}({\mathbf L}^{{\mathbf X}})\), \({\mathbf a}\leq{\mathbf f}({\mathbf a})\); the set of all \({\mathbf R}\)-maps on \({\mathbf L}^{{\mathbf X}}\) is denoted by \({\mathfrak R}({\mathbf L}^{{\mathbf X}})\). A pointwise uniformity on \({\mathbf L}^{{\mathbf X}}\) is a nonempty subset \({\mathcal U}\) of \({\mathfrak R}({\mathbf L}^{{\mathbf X}})\) satisfying certain conditions. This paper contains some characterizations of pointwise uniformities on \({\mathbf L}^{{\mathbf X}}\).
A \({\mathbf T}_2\)-axiom compatible with the pointwise metric is defined and the expected relations with normality, pointwise complete regularity, regularity and \({\mathbf T}_1\) are proved. The Urysohn metrization theorem and Alexandroff-Urysohn metrization theorems from general topology are generalized to \({\mathbf L}\)-topology.

MSC:
54A40 Fuzzy topology
54E35 Metric spaces, metrizability
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