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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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