Pairwise semi compact and pairwise semi lindeloff spaces

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					   Scientia Magna
  Vol. 5 (2009), No. 3, 62-71




       Pairwise semi compact and pairwise semi
                   lindeloff spaces
                                   Srinivasa Balasubramanian


            Government Arts College (Autonomous), Karur-639 005 (T.N.)-INDIA
                            E-mail: mani55682@rediffmail.com
    Abstract In this paper (i, j) semi compact and pairwise semi compact; (i, j) semi Lindeloff
    and pairwise semi Lindeloff Bitopological spaces are defined and their basic properties are
    studied.

    Keywords Bitopological space, semi open set, (i, j) semi compact, pairwise semi compact,
    (i, j) semi Lindeloff, pairwise semi Lindeloff Bitopological space.



§1. Introduction
      J.C. Kelly[2] introduced the concept of Bitopological spaces in 1963 which paved way to
the theory of Bitopological spaces. After him many Authors defined different version of Bitopo-
logical spaces and many of their properties like, compactness, connectedness, countablity and
separation properties were studied with respect to different type of open sets namely semi open,
pre open and semi pre (β) open sets. Norman Levine[5] introduced the concept of semi open
sets and semi continuity in topological spaces, Maheswari[4] and prasad[4] extended the notions
of semi open sets and semi continuity to Bitopological spaces. Shantha Bose[6] further inves-
tigated the properties of semi open sets and semi continuity in Bitopological spaces. Ian E.
Cooke and Ivan L. Reilly[8] defined and studied the basic properties of compactness in Bitopo-
logical spaces. F. H. Khedr et.al.[3] studied interrelations between different open sets between
Bitopological spaces. Recently S. Balasubramanian and G. Koteeswara Rao introduced weak
and strong Lindeloff Bitopological spaces. In this paper the author introduced compactness
using semi open sets in Bitopological spaces which is independent of compactness defined by
others and tried to extend the concepts of Lindeloff condition and discussed basic properties in
(i,j) and pairwise semi lindeloff spaces.


§2. Preliminaries
    A non empty set X together with two topologies τ1 and τ2 is called a Bitopological space[2].
Hereafter a space X is called as a Bitopological space unless otherwise stated in this paper. A
subset A of X is called (τi , τj ) semi open (briefly (i, j) semi open)[3] if there exists U ∈ τi such
that U ⊂ A ⊂ Clj (U ). A subset A is said to be pairwise semi open if it is (i, j) semi open
and (j, i) semi open. A space X is called a pairwise compact[8] if every pairwise open cover
Vol. 5                   Pairwise semi compact and pairwise semi lindeloff spaces                       63

U ∈ τi ∪ τj has a finite sub cover. A space is called a weak (strong) [locally] compact if it is
either τi or τj (τi and τj ) [locally] compact.
     Definition 2.1.[6] A subset A of a topological space (X, τ ) is said to be a semi open set
if there is an open set U 
				
DOCUMENT INFO
Description: Proof. Let A be (i,j) semi open in Y, then there exists W ∈ σ^sub i^ such that W ⊂ V ⊂ cl^sub j^(W). Since f is pairwise open, it follows that f^sup -1^(W) ⊂ f^sup -1^ (V) ⊂ f^sup -1^(cl^sub j^(W)) ⊂ cl^sub j^(f^sup -1^ (W)). Since f is pairwise semi continuous, f^sup -1^(W) is (i,j) semi open in X. By theorem 2.5, f^sup -1^(V) is (i, j) semi open in X.
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