Rings of continuous functions and nonlinear partial differential equations

Abstract

This work consists of two parts. The first deals mainly with the relationships between the algebraic and order structures of the rings of continuous and bounded continuous real-valued functions on a topological space, and the topological structure of the space itself. The second is concerned with the application of both the algebraic and order structures of rings of continuous functions in solving nonlinear partial differential equations (PDE's). Section One is an introduction to compactifications and completely regular spaces, which are precisely the spaces that have compactifications, a consequence of Tychonoff's Characterization Theorem. Completely regular spaces are also the largest class of topological spaces whose rings of continuous functions need be studied by means of compactifications, since these spaces are not distinguishable from more general spaces through the algebraic and lattice properties of their rings of continuous functions, as proved by E. Cech and M. H. Stone. The Stone-Cech compactification of a completely regular space, which is characterized by the property that all bounded continuous functions on that space can be continuously extended to the compactification, is studied in the second section. It is shown to be the unique largest possible compactification, every other compactification being a retract of it. A study is made of the relationships between the topological structures of completely regular spaces, their Stone-Cech compactifications, the algebraic structures of their rings of continuous and bounded continuous functions, and the topological structures of the maximal ideal spaces of these rings, culminating in the Gelfand-Kolmogoroff Theorem. Section Three deals with nonmeasurable cardinals and discrete realcompact spaces. Realcompactness is a topological property intimately linked to Gelfand and Kolmogoroff's characterization of the maximal ideals of rings of continuous functions in terms of the points of the Stone-Cech compactification. A characterization is given for the realcompactness of a discrete space in terms of the nonmeasurability of the cardinality of the space, and the class of nonmeasurable cardinals is shown to be closed with respect to the operations of cardinal arithmetic. In the fourth section, a simple algebraic characterization of the existence of generalized solutions for certain continuous polynomial nonlinear PDE's is obtained, in terms of a simple neutrix, or off-diagonal, condition on certain ideals of subrings of rings of sequences of continuous functions. The spaces of generalized functions used are quotient vector spaces and quotient rings constructed from these ideals and subrings. An order completion generalized solution method for similar nonlinear PDE's is developed in the fifth section. In this case, after first constructing polynomials which provide sufficiently many local classical subsolutions, unique global generalized solutions are constructed through the Dedekind order completion of spaces of generalized functions patched up from locally smooth functions. Six appendices discuss the algebraic and order structures of the function rings, the basic theory of z-filters, the hull-kernel topology of maximal isome background on cardinals, and order structure and the basic Dedekind order completion of a partially ordered set as given by Mac Neille's construction.