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Introduction
Bitopological space , initiated by Kelly [ 7 ], is by definition a set equipped with two non identical topologies , and it is denoted by (X, ?,?) where ? and ? are two topologies defined on X .Bitopological space , initiated by Kelly [ 7 ], is by definition a set equipped with two non identical topologies , and it is denoted by (X, ?,?) where ? and ? are two topologies defined on X .
A sub set F of a topological space (X , ?) is F? [ 11 ] if it is a countable union of ?-closed set . We will denote to such set by ?- F ? .
Let (X , ?) be a topological space . A cover (or covering) [ 3 ] of a space X is a collection of subset of X whose union is the whole X .
A sub cover of a cover U [ 3 ] is a sub collection v of u which is a cover .An open cover of X [ 3 ] is a cover consisting of open sets , and other adjectives appling to subsets of X apply similarly to covers .
For an infinite cardinal number m , if the collection consists of at most m sub-sets, we say that it has cardinality or simply card. . Some times this collection is denoted by .
If a sub set A of X is consisting of at most m elements we say that A has cardinality (or with cardinality ) , and is denoted by . A bitopo-logical space (X, ?, ?) is called (m) (?-?) compact if for every ?-open cover of X, (with cardinality ), it has ?-open sub-covers .
The function is said to be func-tion if the image [inverse image of each ?-closed[?`-open ] is ?`-closed [?- open in X] in Y. .Let U={U? : ???} and V={V? :???} be two coverings of X , V is said to be refine (or to be a refinement of ) U , if for each V? there exists some U? with V? U? .
If W={W? :? ?} refine two covers U, V of X, then it is called common refinement [2] . A family U={ U? : ? ?}of sets in a space (X,?) is called local- ly finite, if each point of X has a neighborhood V such that V?U?? for at most finitely many indices ?. In other word V?U?= for all but a finite number of ?. A family U of set in a space (X,?) is called ?-locally finite if where each Un is a locally finite collection in X.
A bitopological space (X, ? ,?) is called pairwise Hausdorff if for every two distinict points x and y of X, there exist ? -open set U and a µ -open set V such that x ? U, y ? V and U ?V= .
A bitopological space (X, ? ,?) is called (m)(?, ? , ?)- regular if for every point x in X and every ? -closed set A with such that for x ? A, there exist two ?- open sets U, V such that x ? U, A V, and U ?V= .Clearly every (?, ?, ?)-regular space is m(?,?,?)-regular space .
A bitopological space (X,?,?) is called (m-)(?,?,?) –normal if for every pair disjoint ?-closed sets A,B of X,with there exist two ?- open sets U,V such that A U, B V, and U?V = .
Clearly every (?,?,?) –normal space is m(?,?,?) –normal.
A topological space (X , ? )is said to be : m-paracompact [9], if every open cover of X with card .?m has a locally finite open refinement.
paracompact[4], if every open cover of X has a locally finite open refinement.
(m-) semiparacompact, if every open cover of X ( with card. ?m) has a ?-locally finite open refinement .
(m-) a-paracompact[1] if every open cover of X with card. ?m has a - locally finite refinement not necessary either open or closed.
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- On paracompact in bitopological spaces
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