# 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 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 CompactopenTopology 2013-03-22 13:25:26 2013-03-22 13:25:26 antonio (1116) antonio (1116) 8 antonio (1116) Definition msc 54-00 topology of compact convergence UniformConvergence