Titlebook: Rings of Continuous Functions; Leonard Gillman,Meyer Jerison Textbook 1960 Springer-Verlag New York 1960 Compactification.algebra.class.fu

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Discrete Spaces. Nonmeasurable Cardinals,alcompact. The question arises whether . discrete spaces are realcompact. Since, among discrete spaces, the cardinal is the only significant variable, this is, in fact, a question about cardinal numbers.

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Hyper-Real Residue Class Fields,). Although none of the material developed after Chapter 5 will be called upon, we shall need quite a bit more of the abstract theory of fields than heretofore. We begin with a summary of these algebraic prerequisites.

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Prime Ideals,heorem 5.5), and hence that the canonical homomorphism .) of . onto . is a lattice homomorphism as well. Moreover, the integral domain . is totally ordered. The set of images of the constant functions is a copy of ., and we identify this copy with . itself.

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Uniform Spaces,y realcompact space admits a complete structure. One of the outstanding successes of the theory of rings of continuous functions is Shirota’s result that, barring measurable cardinals, the converse is also true, so that the spaces admitting complete structures are precisely the realcompact spaces.

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Ordered Residue Class Rings,pter initiates the study of residue class fields modulo arbitrary maximal ideals. Each such field has the following properties, as will be shown: it is a totally ordered field, whose order is induced by the partial order in ., and the image of the set of constant functions is an isomorphic copy—nece