# 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

$${\mathcal{U}}_{K,U}:=\{f\in C(X,Y):f(x)\in U\text{whenever}x\in K\}.$$ |

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

$$\{{\mathcal{U}}_{K,U}:K\subset X\text{compact,}\mathit{\hspace{1em}}U\subset Y\text{open}\}.$$ |

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 |