C(X) determines X - an inherent theory

Abstract

One of the fundamental problem in rings of continuous function is to extract those spaces for which C(X) determines X, that is to  investigate X and Y such that C(X) isomorphic with C(Y ) implies X homeomorphic with Y. The development started back from Tychonoff who first pointed out inevitability of Tychonoff space in this category of problem. Later S. Banach and M. Stone proved independently with slight variance, that if X is compact Hausdorff space, C(X) also determine X. Their works were maximally extended by E. Hewitt by introducing realcompact spaces and later Melvin Henriksen and Biswajit Mitra solved the problem for locally compact and nearly realcompact spaces. In this paper we tried to develop an inherent theory of this problem to cover up all the works in the literature introducing a notion so called P-compact spaces.