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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.)
##### Keywords:
cozero set; $$\omega$$-open set; Lindelöf; $$z$$-Lindelöf
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