Definition:Strongly Locally Compact Space

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Definition 1

The space $T$ is strongly locally compact if and only if:

every point of $S$ is contained in an open set whose closure is compact.

Definition 2

The space $T$ is strongly locally compact if and only if:

every point has a closed compact neighborhood.

That is:

every point of $S$ is contained in an open set which is contained in a closed compact subspace.

Also defined as

Some sources define a strongly locally compact space as what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is called a locally compact space.

Also see

• Equivalence of Definitions of Strongly Locally Compact Space

• Definition:Locally Compact Space
• Definition:Weakly Locally Compact Space

• Strongly Locally Compact Space may not be Locally Compact
• Locally Compact Space may not be Strongly Locally Compact
• Sequence of Implications of Local Compactness Properties

• Results about strongly locally compact spaces can be found here.