# nLab compact-open topology

### Context

#### Topology

topology

algebraic topology

mapping space

# Contents

## Idea

A natural topology on mapping spaces of continuous functions.

## Definition

Let $X$ and $Y$ be topological spaces. The set $\mathrm{Map}\left(X,Y\right)$ (often denoted also $C\left(X,Y\right)$) of continuous maps from $X$ to $Y$ has a natural topology called the compact-open topology: a subbase of that topology consists of sets of the form ${U}_{K,V}$, where $K\subset X$ is compact and $V\subset Y$ is open?, which consists of all continuous maps $f:X\to Y$ such that $f\left(K\right)\subset V$.

If $Y$ is a metric space then the compact-open topology is the topology of uniform convergence on compact subsets in the sense that ${f}_{n}\to f$ in $\mathrm{Map}\left(X,Y\right)$ with the compact-open topology iff for every compact subset $K\subset X$, ${f}_{n}\to f$ uniformly on $K$. If (in addition) the domain $X$ is compact then this is the topology of uniform convergence.

## Properties

The compact-open topology is most sensible when the topology of $X$ is locally compact Hausdorff, for in this case $\mathrm{Map}\left(X,Y\right)$ with the compact-open topology is an exponential object ${Y}^{X}$ in the category Top of all topological spaces. This implies the exponential law for spaces , i.e. the adjunction map is a bijection $\mathrm{Top}\left(X,\mathrm{Map}\left(Y,Z\right)\right)\cong \mathrm{Top}\left(X×Y,Z\right)$ whenever $Y$ is locally compact Hausdorff; and it becomes a homeomorphism $\mathrm{Map}\left(X,\mathrm{Map}\left(Y,Z\right)\right)\cong \mathrm{Map}\left(X×Y,Z\right)$ if in addition $X$ is also Hausdorff. See also convenient category of topological spaces.

Revised on June 6, 2011 06:30:04 by Tim Porter (95.147.238.100)