Stone-Cech compactification and rings of continuous functions

Abstract

The overall purpose of this thesis is to define and to investigate rings of continuous functions defined on completely regular topological spaces. One of the objectives, achieved in Chapter 1 and Chapter 3, is to present and discuss relationships between topological properties of a space X and algebraic properties of its corresponding rings of continuous functions. Both rings of continuous functions considered in the thesis (namely C(X) - the ring of continuous functions, and c*(X) - the ring of bounded continuous functions) are uniquely determined by the space X. Thus, it is natural to examine the converse of this fact, that is, the specification of conditions under which the space X is determined by the algebraic structure of C(X) or that of c*(X). The theory developed in this thesis will build up to show that, within the class of compact spaces, the ring structure of c*(X) determines the space X up to homeomorphism; in other words, the ring c* distinguishes among compact spaces. Analogous results will be also proved for the ring C and realcompact spaces. Another interesting aspect of the theory of rings of continuous functions, presented in Chapter 1, is the fact that several important properties of the continuous functions on a space X (like order structure and boundedness of functions) are determined by the ring structures of C(X) and c*(X). The relationship between C---embeddings and c* -embeddings of various topological spaces is also established therein. Chapter 2 deals with various types of compactifications and methods of compactifying topologic spaces. Its main purpose is to study the Stone-Cech compactification which, from the point of view of this thesis, is the most important and interesting type of compactification. It is shown that such a compactification exists for every completely regular space; that it is the "largest" compactification and that it is unique. Several of its characteristics are investigated; as well as its use in determining of the relationships between a space and its rings of continuous functions and their sets of maximal ideals. Finally, various techniques of constructing the Stone-Cech compactification are discussed, accompanied by examples thereof. In Chapter 4, an interesting application of the theory of rings of continuous functions is discussed. The chapter presents a global version of the well-known Cauchy-Kovalevskaia theorem for nonlinear partial differential equations (PD Es). Its goal is to prove the existence of global generalized solutions for arbitrary analytic nonlinear PDEs on the whole of their domains of analyticity, and shows that these solutions are analytic outside of closed, nowhere dense subsets. One global and universal principle which can define sets of "patched up" solutions for arbitrary analytic nonlinear PDEs is also presented in this chapter. The proofs are based on constructions within rings of continuous functions on Euclidean spaces.