The concept of Paracompactness is due to Dieudonne [6] . The concept of paracompact with respect to
three topologies is due to Martin [5] . The term space ( X ,t ,? ) is referred to as a set X with two
generally nonidentical topologies t and ? .
A cover ( or covering ) of a space ( X , t ) is a collection of subsets of X whose union is all of
X . A t -open cover of X is a cover consisting of t -open sets , and other adjectives applying to subsets
of X apply similarly to covers . If C and ? are covers of X , we say ? refines C if each members
of ? is contained in some member of C . Then, we say? refines ( or is a refinement of ) C . A
collection ? of subsets of X is called locally finite if each x in X has a neighborhood meeting only
finitely many member of ? , and is called s -locally finite if it is a countable union of locally finite
collection in X . Note that , every locally finite collection of sets is s -locally finite . A subset of a
topological space ( X , t ) is an Fs if it is a countable union of t - closed sets , and written by t - F