Locally compact space

From Topospaces
Jump to: navigation, search
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 \ni x
  • For every point x \in X, there exists an open subset U \ni x, such that \overline{U} is compact
  • For every point x \in X, there exists an open subset U \ni x, and a closed compact subset K of X such that U \subset K

Relation with other properties

Stronger properties

Weaker properties

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