Locally compact space

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 $$U \subset K$$

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
Any closed subspace of a locally compact space is locally compact.