عنوان البحث(Papers / Research Title)
On Semiparacompactness and z-paracompactness in
الناشر \ المحرر \ الكاتب (Author / Editor / Publisher)
لؤي عبد الهاني جبار السويدي
Citation Information
لؤي,عبد,الهاني,جبار,السويدي ,On Semiparacompactness and z-paracompactness in , Time 5/8/2011 9:41:20 AM : كلية التربية للعلوم الصرفة
وصف الابستركت (Abstract)
On Semiparacompactness and z-paracompactness in
الوصف الكامل (Full Abstract)
SummaryWe find some properties of semi paracompactness and z-paracompactness inbitopological spaces and give the relation between these concepts. Throughout the presentpaper m will denote infinite cardinal numbers.Keywords: Paracompact, z-paracompact and bitopological spaces
1. Introduction
The concept of Paracompactness is due to Dieudonne [6] . The concept of paracompact with respect tothree topologies is due to Martin [5] . The term space ( X ,t ,? ) is referred to as a set X with twogenerally nonidentical topologies t and ? .A cover ( or covering ) of a space ( X , t ) is a collection of subsets of X whose union is all ofX . A t -open cover of X is a cover consisting of t -open sets , and other adjectives applying to subsetsof X apply similarly to covers . If C and ? are covers of X , we say ? refines C if each membersof ? is contained in some member of C . Then, we say? refines ( or is a refinement of ) C . Acollection ? of subsets of X is called locally finite if each x in X has a neighborhood meeting onlyfinitely many member of ? , and is called s -locally finite if it is a countable union of locally finitecollection in X . Note that , every locally finite collection of sets is s -locally finite . A subset of atopological space ( X , t ) is an Fs if it is a countable union of t - closed sets , and written by t - Fs .
1.1. Lemma [6]Let U be a cover of a topological space X , and let V be a refinement of U . If W refines V , then Wrefines U .1.2. Lemma [6]Let (Y , Y t ) be a subspace of ( X , t ). If a collection = { :g IG} g V V of sets is a (s -)locally finitewith respect to t , then so is { C :g I G} g V Y with respect to Y t .On Semiparacompactness and z-paracompactness in Bitopological Spaces 5551.3. Lemma [6]1. If = { :l ID} l U U is locally finite collection of sets in( X , t ) . Then any subcollection of U is locally finite .2. If = { :l ID} l U U is locally finite collection of sets in( X , t ) , then so is { ( ) :l ID} t l cl U and U ( ) (U )ID ID=ll t llt cl U cl U .3. The union of a finite number of locally finites collections of sets is locally finite.
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