Separation axioms: Extension of mappings and embedding of spaces (Q2702346)

This paper is an interesting and systematic development of some of the topics in fuzzy topology that involve \(L\)-real-valued, i.e., \(R(L)\) (the \(L\)-real line) valued, functions on \(L\)-topological spaces. The main themes include separation of \(L\)-set by \(L\)-real functions, insertion of \(L\)-real continuous functions, complete \(L\)-regularity and \(L\)-normality of \(L\)-topological spaces, etc. It is shown that most of the relevant fundamental results in general topology such as the Uryson lemma, the Tietze extension theorem and the Tikhonov embedding theorem, etc, can be generalized to the \(L\)-setting, here \(L\) stands for a complete lattice (without any assumption of distributivity) in most cases. However, since the assumption of \(L\) has been minimalized, the author does not touch on one of the most important aspects of complete \(L\)-regular spaces: \(L\)-uniformizability of these spaces. It is known if \(L\) is completely distributive, then an \(L\)-topological space is completely \(L\)-regular if and only if it is uniformizable in the sense of Hutton.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].