Equivalence of Definitions of Strongly Locally Compact Space

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

The following definitions of the concept of Strongly Locally Compact Space are equivalent:

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.

Proof

1 implies 2

Follows immediately from Topological Closure is Closed.

$\Box$

2 implies 1

Let $x \in S$

Let $U$ be an open neighborhood of $x$.

Let $K \subseteq S$ be a closed compact subspace with $U \subseteq K$.

By Topological Closure is Closed $\overline U$ is closed in $T$.

By Set Closure as Intersection of Closed Sets, the closure $\overline U$ of $U$ in $T$ satisfies $\overline U \subseteq K$.

By Closed Set in Closed Subspace, $\overline U$ is closed in $K$.

By Closed Subspace of Compact Space is Compact, $\overline U$ is compact.

$\blacksquare$