compact-open topology

Let $X$ and $Y$ be topological spaces, and let $C(X,Y)$ be the set of continuous maps from $X$ to $Y .$ Given a compact subspace $K$ of $X$ and an open set $U$ in $Y,$ let

{\mathcal{U}}_{K,U}:=\left\{f\in C(X,Y):\>f(x)\in U\,\text{whenever}\,x\in K% \right\}.

Define the compact-open topology on $C(X,Y)$ to be the topology generated by the subbasis

\left\{{\mathcal{U}}_{K,U}:\>K\subset X\,\text{compact,}\quad U\subset Y\,% \text{open}\right\}.

If $Y$ is a uniform space (for example, if $Y$ is a metric space), then this is the topology of uniform convergence on compact sets. That is, a sequence $\left(f_{n}\right)$ converges to $f$ in the compact-open topology if and only if for every compact subspace $K$ of $X,$ $\left(f_{n}\right)$ converges to $f$ uniformly on $K$ . If in addition $X$ is a compact space, then this is the topology of uniform convergence.

Title	compact-open topology
Canonical name	CompactopenTopology
Date of creation	2013-03-22 13:25:26
Last modified on	2013-03-22 13:25:26
Owner	antonio (1116)
Last modified by	antonio (1116)
Numerical id	8
Author	antonio (1116)
Entry type	Definition
Classification	msc 54-00
Synonym	topology of compact convergence
Related topic	UniformConvergence