Some Topological Properties of I(X)-Spaces

Abstract

The set of all isometries on a metric space X with the usual composition of functions forms a group and it is called the group of isometries and is denoted by I(X).In this work we illustrate that this group forms a topological transformation group with pointwise convergence topology. We also give relationships between the compactness of X and I(X).From the main goals of our work is searching about a necessary condition which make the group of isometries locally compact. We find that the limit sets have active situation for suggestion, so we study the closeness of limit sets