Concerning rings of continuous functions

Abstract

The present paper deals with two distinct, though related, questions, concerning the ring C(X, R) of all continuous real-valued functions on a completely regular topological space X. The first of these, treated in ? ?1-7, is the study of what we call P-spacesthose spaces X such that every prime ideal of the ring C(X, R) is a maximal ideal. The background and motivation for this problem are set forth in ?1. The results consist of a number of theorems concerning prime ideals of the ring C(X, R) in general, as well as a series of characterizations of P-spaces in particular. The second problem, discussed in ??8-10, is an investigation of what Hewitt has termed Q-spaces-those spaces X that cannot be imbedded as a dense subset of any larger space over which every function in C(X, R) can be continuously extended. An introduction to this question is furnished in ?8. Our discussion of Q-spaces is confined to the class of linearly ordered spaces (introduced in ?6). We are able to settle the question as to when an arbitrary linearly ordered space is or is not a Q-space. The concept of a paracompact space turns out to be intimately related to these considerations. We also derive a characterization of linearly ordered paracompact spaces, and we find in particular that every linearly ordered Q-space is paracompact. A result obtained along the way is that every linearly ordered space is countably paracompact.