Locally Hausdorff space

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 Hausdorffness. View other variations of Hausdorffness

Definition

Symbol-free definition

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

• Every point has an open neighbourhood which is Hausdorff
• Given any point, and any open neighbourhood of the point, there is a smaller open neighbourhood of the point which is Hausdorff.

Relation with other properties

Stronger properties

• Hausdorff space: For proof of the implication, refer Hausdorff implies locally Hausdorff and for proof of its strictness (i.e. the reverse implication being false) refer locally Hausdorff not implies Hausdorff
• Locally metrizable space
• Locally Euclidean space

Weaker properties

• T1 space

Metaproperties

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Products

This property of topological spaces is closed under taking finite products

Local nature

This property of topological spaces is local, in the sense that the topological space satisfies the property if and only if every point has an open neighbourhood which satisfies the property