# characterization of compactly generated space

We give equivalent^{} conditions when a Hausdorff topological space is compactly generated. First we need a definition.

Definition. Let $X$ be a ${T}_{1}$ space and $Y$ a topological space^{}. We define ${C}_{k}(X,Y)$ to be the set of functions $f:X\to Y$ such that for every compact^{}, closed set^{} $K\subset X$ the restriction^{} ${f}_{|K}$ is continuous^{}.

Clearly, $C(X,Y)\subset {C}_{k}(X,Y)$ for such spaces $X,Y$. With this we have the following theorem.

Theorem. Let $X$ be a Hausdorff space. Then the following conditions are equivalent.

*i)* $X$ is compactly generated

*ii)* $X$ carries the final topology generated by the family of inclusion mappings ${({i}_{K}:K\to X)}_{K\subset X\text{compact}}$.

*iii)* For every topological space $Y$ we have ${C}_{k}(X,Y)=C(X,Y)$.

*iv)* $X$ is an image of a locally compact space under a quotient mapping.

Remark. It follows easily from this that if there is a quotient mapping $f:X\to Y$ which maps a compactly generated space $X$ onto a Hausdorff space $Y$ then $Y$ is compactly generated.

Title | characterization of compactly generated space |
---|---|

Canonical name | CharacterizationOfCompactlyGeneratedSpace |

Date of creation | 2013-03-22 19:10:04 |

Last modified on | 2013-03-22 19:10:04 |

Owner | karstenb (16623) |

Last modified by | karstenb (16623) |

Numerical id | 4 |

Author | karstenb (16623) |

Entry type | Theorem |

Classification | msc 54E99 |

Defines | C_k(X |

Defines | Y) |