General rings of functions

Abstract

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

Inserting L‐fuzzy Real‐valued Functions

“…The terminology which follows is similar to that of SIKORSKI [27] (cf. [29] An inspection of our proofs shows that these were given without involving uncountable operations in Lx, in particular the closure. In other words, the results remain valid if one replaces a normal L-topology by a normal o-ring in Lx (with measurability instead of continuity).…”

“…This kind of generalization is well-known in general topology. For instance, normal O-rings of subsets have been characterized in terms of a Urysohn's type lemma in [t], [14] and [28] (see also [29] and [13]). In [15], the other characterizations of normality by real-valued functions have been extended to normal O-rings of subsets.…”