Paracompactness of Extremally Disconnected Spaces

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Abstract


In this paper we consider the ω-mappings defined by semi-open sets, i.e. sets that are unions of open sets and subsets of their boundaries. This are quasicontinuous ω-mappings. (The mapping f : X → Y is called quasi-continuous if for any open set G ⊂ Y the set f-1(G) is a semi-open set). Characterization of paracompactness based on continuous ω-mappings is well known. Of interest is the question of to what extent it is possible to waive the requirement of continuity of ω-mappings in the characterization of paracompactness of topological spaces with those, or other additional properties. One of these properties is extreme disconnectness. The main goal of our work is to characterize extremely disconnected paracompact space by ω-mapping on the metric space, loosening the requirement of continuity. We have proved that extremely disconnected space X is paracompact if and only if for any open covering ω of X there exists a quasi-continuous ω-mapping on some metric space.

About the authors

V L Klyushin

Peoples’ Friendship University of Russia

Email: vklyushin@mail.ru
Higher Mathematics Departement

- Jalal Hatem Hussein Al Bayati

Peoples’ Friendship University of Russia

Email: jalalintuch@yahoo.com
Higher Mathematics Departement

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Copyright (c) 2015 Клюшин В.Л., Джелал Хатем Хуссейн Аль Баяти -.

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