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