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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.
MSC:
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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