We develop new results for partial stability of general dynamical systems with respect to invariant sets defined on metric space, using stability preserving mappings. Our results are applicable to a much larger class of systems than existing results, including to dynamical systems that cannot be determined by the usual classical (differential) equations. Furthermore, in contrast to existing results which pertain primarily to the analysis of equilibria, the present results apply to invariant sets (including equilibria as a special case). We apply our results in the analysis of a class of discrete event systems (a computer load balancing problem).