Locally compact 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 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

• 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