Let X be a compact space, (Y,p) a compact metric space, let F be a closed subset of C(X,Y) (the continuous functions space) (i guess it obviously means in the open-compact topology, although it's not mentioned there) which satisifes:(adsbygoogle = window.adsbygoogle || []).push({});

for every e>0 and every x in X there exists a neighbourhood U of x s.t for every f in F:

f(U) is a subset of B_e(f(x)) which is an open ball with radius e.

prove that F is compact.

well i want to prove that C(X,Y) is compact and then it will follow that F is compact.

now let U(W(C,U))=C(X,Y) which is an arbitrary covering of C(X,Y), i.e for every C compact in X and every U open in Y s.t f(C) is contained in U.

now i need to find a finite covering, now because Y is compact and metric then it is covered by a finite union of open balls, because f in F is continuous then f^-1(B_e(f(x))

is open in X, so finite union of them cover the entire set X, so because every compact set in X is actually covered by a finite union of f^-1(B_e(f(x))) and also every every open set in Y is covered by a finite union of open balls, then C(X,Y) is covered by a finite union of W(C_i,B_e_j(f(x))) s.t C_i is compact in some f^-1(B_e_j(f(x)) i.e f(C_i) is contianed in B_e_j(f(x))), so C(X,Y) is compact and then also F is compact as beiing a closed subset of C(X,Y).

am i right or way off?

thanks in advance.

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# The space of continuous functions.

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