compact-open topology - MathOverflow

compact-open topology

2010-10-31T17:49:45Z

Is there a natural reason for defining the compact-open topology on the set of continuous functions between two locally compact spaces. For example "to make ... functions continuous". Or in another way of asking this, is there an adjoint functor of the functor, say F, which assigns the topological space $F(X,Y):=Hom_C(X,Y)$ (with the compact-open topology on it) to the couple X,Y.

Answer by Peter LeFanu Lumsdaine for compact-open topology

2010-10-31T18:35:28Z

Exactly as you say, adjoint functors are the answer! (Or at least, they're one possible answer.) In particular, for reasonable spaces $X,Y,Z$, there is a natural isomorphism

$\mathrm{Hom} (X \times Y, Z) \cong \mathrm{Hom} (X, [Y,Z])$

where $[Y,Z]$ denotes $\mathrm{Hom}(Y,Z)$ with the compact-open topology. This is exactly the categorical characterisation of an exponential object.

This certainly holds when $X,Y,Z$ are compactly-generated Hausdorff spaces, so the category of such spaces is Cartesian closed. It actually also holds under rather weaker conditions on $X,Y$ and $Z$ individually — I can't recall them off the top of my head though and am in a bit of a rush, so am wiki'ing this answer in hopes someone can fill them in. Now, this also holds if we take weaker conditions on $X$ and $Y$ (just Hausdorffness), and a slightly stronger condition on $Y$ (local compactness). $Z$ may be arbitrary.

Re Mariano's comment: yes, in some sense this is just fancy language for “things we want to converge, converge; things we don't, don't”. But I think this helps explain why we want the things we want to converge, to converge. ☺

Answer by Andrej Bauer for compact-open topology

2010-10-31T19:00:01Z

In regard to your question I recommend Topologies on spaces of continuous functions, Topology Proceedings, volume 26, number 2, pp. 545-564, 2001-2002 by Martin Escardo and Reinhold Heckmann.

Answer by Dylan Wilson for compact-open topology

2010-10-31T19:29:52Z

I always liked the following reason:

Let's call a topology on a space "admissible" if the evaluation function $e: Hom(X,Y) \times X \rightarrow Y$ is continuous. Then the compact-open topology is coarser than any other admissible topology. In particular, in any case where the compact-open topology is admissible, it is the smallest possible topology that does this.

EDIT: See comments for some references. I don't claim any originality here :)