Vol. 5 (2009), No. 3, 62-71
Pairwise semi compact and pairwise semi
Government Arts College (Autonomous), Karur-639 005 (T.N.)-INDIA
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
Keywords Bitopological space, semi open set, (i, j) semi compact, pairwise semi compact,
(i, j) semi Lindeloﬀ, pairwise semi Lindeloﬀ Bitopological space.
J.C. Kelly 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 introduced the concept of semi open
sets and semi continuity in topological spaces, Maheswari and prasad extended the notions
of semi open sets and semi continuity to Bitopological spaces. Shantha Bose further inves-
tigated the properties of semi open sets and semi continuity in Bitopological spaces. Ian E.
Cooke and Ivan L. Reilly deﬁned and studied the basic properties of compactness in Bitopo-
logical spaces. F. H. Khedr et.al. 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.
A non empty set X together with two topologies τ1 and τ2 is called a Bitopological space.
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) 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 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. A subset A of a topological space (X, τ ) is said to be a semi open set
if there is an open set U