Re: Compact-open topology

On Jul 7, 4:11 pm, Timothy Murphy <gayle...@xxxxxxxxxx> wrote:
Daniel Grubb wrote:

Given locally compact (and Hausdorff) spaces X,Y
let C(X,Y) be the space of continuous maps X->Y
with the compact-open topology.

Is there some simple condition on X,Y
that will ensure that C(X,Y) is locally compact?
If Y is a uniform space (in particular a metric space),
the Arzela-Ascoli theorem applies. For a set A to be
pre-compact (compact closure) in the compact-open
topology, A needs to be equicontinuous and have pointwise
compact closure. The equicontinuity is your problem.

In particular, if X is not discrete and Y is locally
path connected metric, the basic open sets on the CO topology
are not equicontinuous. Suppose K is compact in X and U is open
in Y. Pick z,w in U distinct. Let e=d(z,w)/2. Let x in X
be non-isolated and suppose that V is any open set around
x. Pick any y in V distinct from x.

By Urysohn's lemma, there is a g:X->[0,1] with g(x)=0 and
g(y)=1. By path connectedness of U, there is a p:[0.1]->U
with p(0)=z and p(1)=w. Then f=pog is in the basic open set
for C(X,Y) defined by K and U and f(x)=z, f(y)=w. This contradicts
the equicontinuity of the basic open set at x.

OK, thanks.

Of course we know that the dual group G* of a locally compact abelian group
is locally compact.
I was wondering if this was a particular case of a larger class
of function spaces known to be locally compact.
Eg is the space of homomorphisms from one locally compact abelian group
to another known to be locally compact?

But the dual group G* is *not* the complete
mapping space, but a very small part of it.
If you are willing to consider *subspaces*
of mapping spaces, there are lots and lots of
locally compact (and even compact, of course)
ones. Even non trivial ones.

-- m
.

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