Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8266Research ArticleParacompactness of Extremally Disconnected SpacesKlyushinV LHigher Mathematics Departementvklyushin@mail.ruJalal Hatem Hussein Al Bayati-Higher Mathematics Departementjalalintuch@yahoo.comPeoples’ Friendship University of Russia15022015271008092016Copyright © 2015,2015In 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.semi-open setquasi-continuous mappingω-mappings-locally ﬁnite systemparacompactnessextremely disconnected spaceполуоткрытое множествоквазинепрерывное отображениеω-отображениеs-локально конечная системапаракомпактностьэкстремально несвязное пространство