C(X) determines X -- an inherent theory

TL;DR

This paper develops a comprehensive inherent theory for the problem of when the ring of continuous functions C(X) uniquely determines the topological space X, extending previous results to a broader class called P-compact spaces.

Contribution

It introduces the notion of P-compact spaces to unify and extend existing results on when C(X) determines X in various classes of spaces.

Findings

Introduces the concept of P-compact spaces.

Unifies previous results on C(X) determining X.

Extends the theory to cover broader classes of spaces.

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 Tychono? who first pointed out inevitability of Tychono? space in this category of problem. Later S.Banach and M. Stone proved independently with slight variance, that if X is compact Hausdor? 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.