Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given topological spaces $X$ and $Y$ the set $C(X,Y)$ of all continuous functions $f:X\to Y$ becomes a topological space with the compact-open topology (that is the topology generated by the sets $C_{K,U}(X,Y)$ of those continuous functions satisfying $f(K)\subseteq U$, where $K$ ranges over the compact subsets of $X$ and $U$ ranges over the open subsets of $Y$).

It is a rather straightforward exercise that if $Y$ is metrizable and $X$ is hemicompact (meaning that there is a countable family of chosen compact sets in it such that every compact set is contained in one of the chosen ones) then the compact-open topology on $C(X,Y)$ is metrizable.

I'm curious to know how reversible this situation is. In more detail:

1) Is there a condition $C$ such that the following is true: If $Y$ is metrizable and the compact-open topology on $C(X,Y)$ is metrizable then $X$ has property $C$?

2) What would be a 'boundary' type example for a non-hemicompact space $X$ and a metrizable space $Y$ such that $C(X,Y)$ is metrizable?

3) What would be a 'boundary' type example for a non-hemicompact space $X$ and a metrizable space $Y$ such that $C(X,Y)$ is not metrizable?

By 'boundary' type examples I mean ones that illustrate the border line of the necessity of the hemicompact condition. So for 2) it would in some sense be a large space, showing how far one can get from being hemicompact yet still have the metrizability of the compact-open topology, while for 3) it would be in some sense a small space.

Thanks!

share|cite|improve this question
up vote 1 down vote accepted

See:

Robert A. McCoy, Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Springer Lecture Notes in Math, Volume 1315 (1988), page 68,

for a metrizable space $Y$, the space of continuous functions $C(X,Y)$ (equipped with the compact-open topology) is metrizable iff $X$ is hemicompact.

In the book, they assume that $X, Y$ are completely regular Hausdorff and that $Y$ contains a nontrivial path, but maybe the latter requirement can be omitted in this theorem. (It is left as an exercise, but, I think, it follows from the prior results in the book.)

Hope it helps.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.