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
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.
|| Statement with symbols
| box product-closed property of topological spaces
|| 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
|| 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
|| (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
|| 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