Characterizations of \(z\)-Lindelöf spaces. (English) Zbl 1424.54051

This article is concerned with certain characterizations and a study of \(z\)-Lindelöf spaces, introduced in 2009 by A. T. Al-Ani, through some new types of sets formulated by the authors; the notion of \(\omega\)-cozero set is the basic one for the purpose. The authors define a subset \(A\) of a topological space \((X,\tau)\) to be \(\omega\)-cozero if to each \(a\in A\) there corresponds a cozero set \(U_{a}\) containing \(a\) such that \(U_{a}-A\) is countable (the complement of an \(\omega\)-cozero set is called an \(\omega\)-zero set). A space \((X,\tau)\) is called \(z\)-Lindelöf if every cover of \(X\) by cozero sets has a countable subcover. Clearly the notion of \(\omega\)-cozero set is a modulated version of cozero set. The relations of \(\omega\)-cozero sets with cozero sets, open sets and \(\omega\)-open sets are presented through diagrams and examples. Further it is demonstrated that for a topological space \((X,\tau)\), the \(\omega\)-cozero sets form a topology on \(X\), just like the cozero sets. Next some characterizations of \(\omega\)-cozero sets are delivered. Then come the main results – formulations of \(z\)-Lindelöf space through \(\omega\)-cozero sets and \(\omega\)-zero sets. After that certain basic results concerning \(z\)-Lindelöfness, analogous to those for Lindelöf spaces, are derived. For this purpose a good many new notions are introduced in the paper. These are completely \(\omega\)-regular space, almost \(\omega\)-regular space, cozero-irresolute function, almost cozero function, \(\omega\)-zero function, \(\omega\)-cozero-continuous function and \(\omega^*\)-cozero-continuous function. These concepts are utilized to study \(z\)-Lindelöf spaces, and their preservation properties for direct and inverse images under the stated types of maps.


54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C05 Continuous maps
54C08 Weak and generalized continuity
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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