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

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

• Every point is contained in a relatively compact open neighbourhood
• 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

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
• Locally Euclidean space

Weaker properties

• Locally paracompact space