Locally Hausdorff space: Difference between revisions

Latest revision as of 20:40, 30 May 2016

Definition

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

For every point $x \in X$ , there is an open subset $U$ of $X$ containing $x$ which is Hausdorff in the subspace topology.
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 $X$ .
$X$ is a union of open subsets each of which is a Hausdorff space with the subspace topology.
$X$ 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 $τ_{1}$ and $τ_{2}$ are two topologies on a set $X$ , such that $τ_{1} \subseteq τ_{2}$ , i.e., every subset of $X$ open with respect to $T_{1}$ is also open with respect to $τ_{2}$ . Then, if $X$ is locally Hausdorff with respect to $τ_{1}$ , it is also locally Hausdorff with respect to $τ_{2}$ .

Relation with other properties

Stronger properties

Property	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

@@ Line 1: / Line 1: @@
+==Definition==
+A [[topological space]] <math>X</math> is termed '''locally Hausdorff''' if it satisfies the following equivalent conditions:
+# For every point <math>x \in X</math>, there is an [[open subset]] <math>U</math> of <math>X</math> containing <math>x</math> which is [[Hausdorff space|Hausdorff]] in the [[subspace topology]].
+# For every point <math>x \in X</math>, and every open subset <math>V</math> of <math>X</matH> containing <math>x</math>, there is an open subset <math>U</math> of <math>X</math> contained in <math>V</math>, and which is [[Hausdorff space|Hausdorff]] in the [[subspace topology]] from <math>X</math>.
+# <math>X</math> is a union of [[open subset]]s each of which is a [[Hausdorff space]] with the [[subspace topology]].
+# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.
 {{topospace property}}
-{{variationof|Hausdorffness}}
+{{variation of|Hausdorff space}}
-==Definition==
+== Formalisms ==
+{{obtained by applying the|locally operator|Hausdorff space}}
-===Symbol-free definition===
+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.
-A [[topological space]] is termed '''locally Hausdorff''' if it satisfies the following equivalent conditions:
+==Metaproperties==
-* Every point has an open neighbourhood which is [[Hausdorff space|Hausdorff]]
+{| class="sortable" border="1"
-* Given any point, and any open neighbourhood of the point, there is a smaller open neighbourhood of the point which is Hausdorff.
+!Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::box product-closed property of topological spaces]] || Yes || [[local Hausdorffness is box product-closed]] || If <math>X_i, i \in I</math> is a (finite or infinite) collection of locally Hausdorff topological spaces, the product of all the <math>X_i</math>s, equipped with the [[box topology]], is also locally Hausdorff.
+|-
+| [[satisfies metaproperty::subspace-hereditary property of topological spaces]] || Yes || [[local Hausdorffness is hereditary]] || Suppose <math>X</math> is a locally Hausdorff space and <math>A</math> is a subset of <math>X</math>. Under the [[subspace topology]], <math>A</math> is also locally Hausdorff.
+|-
+| [[satisfies metaproperty::local property of topological spaces]] || Yes || (by definition) || Suppose <math>X</math> is a locally Hausdorff space and <math>x \in X</math>. Then, there exists an open subset <math>U</math> of <math>X</math> containing <math>x</math> such that <math>U</math> is locally Hausdorff.
+|-
+| [[satisfies metaproperty::refining-preserved property of topological spaces]] || Yes || [[localHausdorffness is refining-preserved]] || Suppose <math>\tau_1</math> and <math>\tau_2</math> are two topologies on a set <math>X</math>, such that <math>\tau_1 \subseteq \tau_2</math>, i.e., every subset of <math>X</math> open with respect to <math>T_1</math> is also open with respect to <math>\tau_2</math>. Then, if <math>X</math> is locally Hausdorff with respect to <math>\tau_1</math>, it is also locally Hausdorff with respect to <math>\tau_2</math>.
+|}
 ==Relation with other properties==
@@ Line 16: / Line 36: @@
 ===Stronger properties===
-* [[Hausdorff space]]
+{| class="sortable" border="1"
-* [[Locally metrizable space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Locally Euclidean space]]
+|-
+| [[Weaker than::Hausdorff space]] || || [[Hausdorff implies locally Hausdorff]] || [[locally Hausdorff not implies Hausdorff]] (the standard example is the [[line with two origins]]) ||{{intermediate notions short|locally Hausdorff space|Hausdorff space}}
+|-
+| [[Weaker than::locally metrizable space]] || || || || {{intermediate notions short|locally Hausdorff space|locally metrizable space}}
+|-
+| [[Weaker than::locally Euclidean space]] || || || ||
+|}
 ===Weaker properties===
-* [[T1 space]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::T1 space]] || points are closed || [[locally Hausdorff implies T1]] || [[T1 not implies locally Hausdorff]] ||
+|-
+| [[Stronger than::Kolmogorov space]] || || || ||
+|}