By John G. Hocking, Gail S. Young

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Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les constructions permettant de parler de limite et de continuité sont d'abord dégagées, puis l'utilité de l. a. compacité pour ramener des problèmes de complexité infinie à l'étude d'un nombre fini de cas est explicitée.

This ebook is the 6th version of the vintage areas of continuing Curvature, first released in 1967, with the former (fifth) variation released in 1984. It illustrates the excessive measure of interaction among staff concept and geometry. The reader will enjoy the very concise remedies of riemannian and pseudo-riemannian manifolds and their curvatures, of the illustration conception of finite teams, and of symptoms of modern growth in discrete subgroups of Lie teams.

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1-11 Function spaces. , by taking subspaces and by making product spaces. A third method, one of particular importance in analysis, is introduced here. This method makes a space out of a collection of functions of one known space into another. We begin the discussion with several well-known examples that present some standard procedures. ExAMPLE 1. Let I dt:>note a closed interval [a, b] in El, and let C(l) be the collection of all real-valued continuous functions defined on I. We topologize C(I) by means of the following metric.

For each a in a, H a n U and H a n V are non empty sets. If the set Ka = H a n (S- (U U V)) were empty, then Ha = (Han U) U (Han V) would be a separation of Ha of the prohibited type. Hence Ka is not empty. Also the sets Ka are simply-ordered by inclusion; for, given any subset X, if H a is contained in H tJ, then H a n X lies in H tJ n X. The subsets Ka therefore satisfy the finite intersection hypothesis and, since S is com- 44 THE ELEMENTS OF POINT-SET TOPOLOGY (CHAP. 2 pact, the intersection naKa is not empty.

AxioM T 3 • If Cis a closed set in the spaceS, and if pis a point not in C, then there are disjoint open sets inS, one containing C and the other containing p. This axiom could be satisfied vacuously if there were no proper closed subsets in the space S. Therefore, in order that there be a large number of closed sets and that we obtain a condition stronger than the Hausdorff, a space is defined to be a T 3 -space if it satisfies both Axiom T 1 and Axiom T 3 . A T 3 -space is usually called a regular space.