From: Robin Chapman
Subject: Re: Function Spaces
Date: Mon, 08 Feb 1999 10:24:59 +1100
Newsgroups: sci.math
To: Felix Dilke
Keywords: Compact-open topology
Felix Dilke wrote:
>
> Can anyone tell me what ways there are to
> topologize the set Hom(X, Y) of all continuous
> maps X->Y, for two topological spaces X & Y?
>
> I'm mainly interested in the case where X, Y are
> compact Hausdorff. It would be nice if Hom(X, Y)
> had the same property.
>
> Please cc replies by e-mail, I can't keep up with this group!
>
> Thanks
>
> Felix Dilke
There are various ways of topologizing the set of
continous maps from one space into another. One of
the most popular is the compact-open topology. If
K is a compact subset of X and U an open subset of Y
let A(K,U) be the set of continuous f:X --> Y with
f(K) contained in U. Then the A(K,U) form a subbasis
for a topology on the function space -- the compact-open
topology.
[I wouldn't use the notation Hom(X,Y) for the function space.
This has conntotations of homomorphism or homeomorphism both
of which are inappropriate here. I'd use C(X,Y) or Map(X,Y).]
--
Robin Chapman + "Going to the chemist in
Department of Mathematics, DICS - Australia can be more
Macquarie University + exciting than going to
NSW 2109, Australia - a nightclub in Wales."
rchapman@mpce.mq.edu.au + Howard Jacobson,
http://www.maths.ex.ac.uk/~rjc/rjc.html - In the Land of Oz