# Locally Hausdorff space

From Topospaces

## Contents

## Definition

A topological space is termed **locally Hausdorff** if it satisfies the following equivalent conditions:

- For every point , there is an open subset of containing which is Hausdorff in the subspace topology.
- For every point , and every open subset of containing , there is an open subset of contained in , and which is Hausdorff in the subspace topology from .
- is a union of open subsets each of which is a Hausdorff space with the subspace topology.
- has a basis comprising Hausdorff spaces.

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of Hausdorff space. View other variations of Hausdorff space

## Formalisms

### In terms of the locally operator

*This property is obtained by applying the locally operator to the property: Hausdorff space*

Note that since Hausdorffness is hereditary, some variants of the locally operator all collapse to the same meaning. In particular, every point being contained in an open Hausdorff subset is equivalent to having a basis of open Hausdorff subsets.

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

box product-closed property of topological spaces | Yes | local Hausdorffness is box product-closed | If is a (finite or infinite) collection of locally Hausdorff topological spaces, the product of all the s, equipped with the box topology, is also locally Hausdorff. |

subspace-hereditary property of topological spaces | Yes | local Hausdorffness is hereditary | Suppose is a locally Hausdorff space and is a subset of . Under the subspace topology, is also locally Hausdorff. |

local property of topological spaces | Yes | (by definition) | Suppose is a locally Hausdorff space and . Then, there exists an open subset of containing such that is locally Hausdorff. |

refining-preserved property of topological spaces | Yes | localHausdorffness is refining-preserved | Suppose and are two topologies on a set , such that , i.e., every subset of open with respect to is also open with respect to . Then, if is locally Hausdorff with respect to , it is also locally Hausdorff with respect to . |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Hausdorff space | Hausdorff implies locally Hausdorff | locally Hausdorff not implies Hausdorff (the standard example is the line with two origins) | |FULL LIST, MORE INFO | |

locally metrizable space | |FULL LIST, MORE INFO | |||

locally Euclidean space |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

T1 space | points are closed | locally Hausdorff implies T1 | T1 not implies locally Hausdorff | |

Kolmogorov space |