# Concerning a weak topology for the space of self maps defined on a metric space

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## Abstract

Mathematicians have done a great deal of work using weak topologies in the study and characterization of various function spaces. This paper concerns results obtained using a weak topology, B[raised *] , on the space of all self maps of a metric space (X,[rho]) . If X is a normed linear vector space the B[raised *]-topology on the space of all linear operators is analogous to the weak-[raised *] topology for the first dual space of X . The B[raised *]-topology is used in characterizing certain subsets of contractions on X which are B[raised *] compact. Each of these subsets forms a topological semigroup under the operation of composition. For complete spaces (X,[rho]) , it will be shown (using the contraction mapping theorem) that these subsets can be written as the disjoint union of compact subsemigroupsb [Xi][lowered x] determined by the points of (X,[rho]) . The next part of the paper deals with a set of linear operators on (X,[rho]) that are B[raised *] -compact where (X,

) is a normed linear space and p(x,y) =

x-y

. The Stone Cech compactification of a completely regular space is then characterized. For compact spaces X, a compactification, Q , of the set of all continuous self maps Q is developed along with the necessary and sufficient conditions that Q be homeomorphic to the Stone Cech compactification [beta]Q .