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Abstract
J. Dieudonne [8], introduced the notions of paracompactness and Martin M. K. [9] , introduced the notions of paracompactness in bitopological spaces and K. AL-Zoubi and S. AL-Ghour [10], introduced the notions of P3-paracompactness of topological space in terms of preopen sets .In this paper, we introduce paracompactness in bitopological spaces in terms of ij-preopen sets . We obtain various characterizations, properties of paracompactness and its relationships with other types of spaces.
Key words: ij-preparacompact, ij-precontinuous, separation axioms .
Introduction
The concepts of regular open , regular closed , semiopen , semiclosed , and preopen sets have been introduced by many authors in a topological space ( cf. [ 1-4] ). These concepts are extended to bitopological spaces by many authors ( cf. [5-7]) .
Throughout the present paper ( X , ) and ( Y, ) ( or simple X and Y ) denote bitopological spaces . when A is a subset of a space X , we shall denote the closure of A and the interior of A in ( X , ) by -clA and -intA , respectively, where i= 1,2 , and i,j = 1,2 ; i j .
A subset A of X is said to be ij- preopen ( resp. ij-semiopen ,ij-regular open , ij-regular closed and ij-preclosed ) if , and . The family of all ij-semiopen ( resp. ij- regular open and ij- preopen ) sets of X is denoted by ij-SO(X) ( resp. ij-RO(X) and ij-PO(X) ) . The intersection of all ij- preclosed sets which contain A is called the ij- preclosure of A and is denoted by ij-PclA . Obviously , ij-PclA is the smallest ij-preclosed set which contains A .
Definition 1.1 .
A bitopological space is called ij-locally indiscrete if every subset of X is .
Definition 1.2 .
A collection of subsets of X is called ,(1) locally finite with respect to the topology ( respectively , ij-strongly locally finite ) , if for each , there exists ( respectively, ) containing x and which intersects at most finitely many members of ;(2) ij-P-locally finite if for each , there exists a ij- preopen set in X containing x and which intersects at most finitely many members of .
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- On preparacompactness in bitopological spaces
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