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By John G. Hocking, Gail S. Young

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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.

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