Algebraic Topology: A Primer (Texts and Readings in by Satya Deo

By Satya Deo

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If Y is locally compact, t h e n t h e c o m p a c t - o p e n topology on C(Y) is a conjoining topology. (b) If p is a compatible uniformity on R, t h e n t h e topology of uniform c o n v e r g e n c e on C(Y) is a conjoining topology. 5. If Y is locally compact, then for any space X, t h e exponential function E is a bijection from C(XxY) o n t o C(X,Ck(Y)). The next theorem establishes the continuity properties of E. is a closed If n e t w o r k on X a n d /3 is a closed n e t w o r k on Y, t h e n define r~x/3 = {A×B : A E ~ a n d B E fl}, which is a closed network on XxY.

D) Ck(X ) is submetrizable. (e) X is almost a - c o m p a c t . 3. p : Ca(X ) ~ The following are equivalent. = C a ( Y ) is a continuous ¢(II{Ca(A): A e ~9}) < 57 (a) Each point of Cp(X) is a G 3 - s e t . (b) Each compact subset of Cp(X) is a G 3 - s e t . (c) Cp(X) has a G$ diagonal. (d) Cp(X)is submetrizable. (e) Cp(X) has coarser separable metrizable topology. 2. 3. 3 can be used to establish a property of spaces having countable networks. 4. continuous image of a separable metric space has a of a base is a network).

5. Let Y be a paraeompact, and let ~ be a closed network on Y. T h e n the following are equivalent. 6. (a) C~(Y) has a splitting topology. (b) /~ is a compact network on Y. (c) For all X and all B E /~, the projection map rrX: X×B -~ X is closed. Fine Topology (cf. 1). (a) If Let (R,p) be a metric space. f: X --* Y is a p e r f e c t map, then the induced function f: cf (Y) P -~ 37 Of (X) is a closed embedding. P (b) If )E is a family HCf (~) is continuous. Z) P In addition, if ~ is finite, then S is a homeomorphism.

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