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 = { l ID} l U U : 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 = { l ID} l U U : consists of
at most m sub-sets, we say that it has cardinality £ m or simply card. £ m . Some
times this collection is denoted by U £ m(or) D £ m .
If a sub set A of X is consisting of at most m elements we say that A has
cardinality £ m (or with cardinality £ m ) , and is denoted by A £ m . A bitopological
space (X,
, ?) is called (m) (
-?) compact if for every
-open cover of
X, (with cardinality £ m), it has ?-open sub-covers . The function
f : (X,t ,?,r )®(Y,t `,?`,r `) is said to be (t ?t `)? close[(t ?t `)continuous] function
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