Pure Mathematics for Theoretical Computer Science PMTCS 2995-3162 10.54216/PMTCS https://www.americaspg.com/journals/show/2510 2023 2023 Mustansiriah University, Department of Mathematics, Iraq Murhaf Obaidi The set of all isometries on a metric space X with the usual composition of functions form a group and it is called the group of isometries and is denoted by I(X). In this paper we study the generalization of the concepts of minimal sets, stability and attraction, from dynamic system into the topological transformation group (I(X),X).We find that the collection of all minimal sets of I(X)-space is the collection of all the closures of orbits of X and we found some useful results about stability and attraction and we fixed the relationship among it's kinds. 2023 2023 30 42 10.54216/PMTCS.020204 https://www.americaspg.com/articleinfo/40/show/2510