Locally Hausdorff space

Definition

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

1. For every point $x\in X$, there is an open subset $U$ of $X$ containing $x$ which is Hausdorff in the subspace topology.
2. For every point $x\in X$, and every open subset $V$ of $X$ containing $x$, there is an open subset $U$ of $X$ contained in $V$, and which is Hausdorff in the subspace topology from $X$.
3. $X$ is a union of open subsets each of which is a Hausdorff space with the subspace topology.
4. $X$ 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 $X_i,i\in I$ is a (finite or infinite) collection of locally Hausdorff topological spaces, the product of all the $X_i$s, equipped with the box topology, is also locally Hausdorff.
subspace-hereditary property of topological spaces	Yes	local Hausdorffness is hereditary	Suppose $X$ is a locally Hausdorff space and $A$ is a subset of $X$. Under the subspace topology, $A$ is also locally Hausdorff.
local property of topological spaces	Yes	(by definition)	Suppose $X$ is a locally Hausdorff space and $x\in X$. Then, there exists an open subset $U$ of $X$ containing $x$ such that $U$ is locally Hausdorff.
refining-preserved property of topological spaces	Yes	localHausdorffness is refining-preserved	Suppose ${\tau }_1$ and ${\tau }_2$ are two topologies on a set $X$, such that ${\tau }_1\subseteq {\tau }_2$, i.e., every subset of $X$ open with respect to $T_1$ is also open with respect to ${\tau }_2$. Then, if $X$ is locally Hausdorff with respect to ${\tau }_1$, it is also locally Hausdorff with respect to ${\tau }_2$.

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