عنوان البحث(Papers / Research Title)
Weak forms of ?-open sets in bitopological spaces and connectedness
الناشر \ المحرر \ الكاتب (Author / Editor / Publisher)
لؤي عبد الهاني جبار السويدي
Citation Information
لؤي,عبد,الهاني,جبار,السويدي ,Weak forms of ?-open sets in bitopological spaces and connectedness , Time 5/8/2011 9:53:34 AM : كلية التربية للعلوم الصرفة
وصف الابستركت (Abstract)
Weak forms of ?-open sets in bitopological spaces and connectedness
الوصف الكامل (Full Abstract)
Abstract:
The aim of this paper is to introduce a new classes of weak ?-open sets in bitopological spaces then study the relations between those classes and some properties . Other aim is to introduce certain type of connectedness in bitopological spaces relative to the new classes of sets introduced in the first part, and get some results .
Keywords: ? pre open set , pre ? open set, ? semi open set, semi ? open set, ? ? open set, ? ? open set, ? ? open set, ? ? open set, ? b open set, b ? open set.
Introduction
The concepts of pre open sets, semi open sets, ? open sets, ? open sets, and b-open sets introduced by many authors in topological spaces (cf. [2, 4, 6, 8, 10] ) and extended to bitopological spaces by others (cf. [9, 11 ] ) . The concept of ?-open sets was introduced and studied by many authors (cf. [ 3,12] ) , and extended to bitopological spaces in [ 5] , by defining the concept of ?1 ?2 –generalized ?-closed set.In this paper many types of weak open sets in bitopological spaces will be defined, Relations between those sets will be discussed, properties such as supra and infra topological structures will be determined.
Also a new type of connectedness for bitopological spaces will be defined and preserving that type of connectedness under certain type of map between bitopological spaces will be proved , many other results and counter examples ,also will be showed.
Throughout this paper the following notation will be used: denotes subset (not necessarily proper), Ac denotes the complement of A in the space (that A is subset of).If ( X, ?1, ?2 ) is a bitopological space, A X, i-int A and j-cl A denote the interior and closure of A relative to ?i and ?j respectively , i-open(closed) set denotes ?i open(closed) set (i,j {1,2}).
Definition [4, 11]
Let ( X, ?1, ?2 ) be a bitopological space, A X, A is said to be : (i) ij- p open set if A i-int (j-clA). (ii) ij- s open set if A j-cl(i-int A). (iii) ij- ? open set if A i-int( j-cl(i-int A)). (iv) ij- ? open set if A j-cl(i-int (j-clA)). (v) ij- b open set if A i-int (j-clA) j-cl(i-int A).( p-open denotes pre open, and s- open denotes semi open) .
1.2 Remark It is clear from definition that in any bittopological space the following hold: (i) Every i-open set is ij- p open , ij- s open, ij- ? open, ij- ? open and ij- b open set. (ii) Every ij- p open set is ij- ? open. (iii) Every ij- ? open set is ij- s open. (iv) Every ij- p open(ij- s open) set is ij- b open set. (v) The concepts of ij- p open and ij- s open sets are independent. (vi) The concepts of ij- ? open and
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