Locally compact space: Difference between revisions

Latest revision as of 19:42, 26 October 2009

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 compactness. View other variations of compactness

Definition

Symbol-free definition

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

Every point is contained in a relatively compact open neighborhood
Every point is contained in an open set, whose closure is a compact subset
Every point is contained in an open set, that is contained in a closed, compact subset

Definition with symbols

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

For every point $x \in X$ , there exists a relatively compact open subset $U ∋ x$
For every point $x \in X$ , there exists an open subset $U ∋ x$ , such that $\bar{U}$ is compact
For every point $x \in X$ , there exists an open subset $U ∋ x$ , and a closed compact subset $K$ of $X$ such that $U \subset K$

Relation with other properties

Stronger properties

Compact space
Strongly locally compact space: Note that this definition coincides with the definition of locally compact if we assume the space is Hausdorff
Locally compact Hausdorff space

Weaker properties

Locally paracompact space

Metaproperties

Weak hereditariness

This property of topological spaces is weakly hereditary or closed subspace-closed; in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property.
View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spaces

Any closed subspace of a locally compact space is locally compact. For full proof, refer: Local compactness is weakly hereditary

Products

This property of topological spaces is closed under taking finite products

@@ Line 1: / Line 1: @@
 {{topospace property}}
-{{variationof|compactness}}
+{{variation of|compactness}}
 ==Definition==
+===Symbol-free definition===
 A [[topological space]] is termed '''locally compact''' if it satisfies the following equivalent conditions:
-* Every point is contained in a [[relatively compact subset|relatively compact]] open neighbourhood
+* Every point is contained in a [[relatively compact subset|relatively compact]] open neighborhood
-* Every point is contained in an open set, which is contained in a compact subset
 * Every point is contained in an open set, whose closure is a compact subset
+* Every point is contained in an open set, that is contained in a closed, compact subset
+===Definition with symbols===
+A [[topological space]] <math>X</math> is termed '''locally compact''' if it satisfies the following equivalent conditions:
-(The equivalence of these follows from the fact that any closed subset of a compact set is compact).
+* For every point <math>x \in X</math>, there exists a relatively compact open subset <math>U \ni x</math>
+* For every point <math>x \in X</math>, there exists an open subset <math>U \ni x</math>, such that <math>\overline{U}</math> is compact
+* For every point <math>x \in X</math>, there exists an open subset <math>U \ni x</math>, and a closed compact subset <math>K</math> of <math>X</math> such that <math>U \subset K</math>
 ==Relation with other properties==
@@ Line 20: / Line 26: @@
 * [[Strongly locally compact space]]: Note that this definition coincides with the definition of locally compact if we assume the space is [[Hausdorff space|Hausdorff]]
 * [[Locally compact Hausdorff space]]
-* [[Locally Euclidean space]]
 ===Weaker properties===
 * [[Locally paracompact space]]
+==Metaproperties==
+{{closed subspace-closed}}
+Any closed subspace of a locally compact space is locally compact. {{proofat|[[Local compactness is weakly hereditary]]}}
+{{finite-DP-closed}}