Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
The following statements are equivalent:
- $(1): \quad T$ is weakly $\sigma$-locally compact
- $(2): \quad T$ is weakly locally compact and Lindelöf
Proof
1 implies 2
Let $T = \struct {S, \tau}$ be weakly $\sigma$-locally compact.
Then by definition:
- $T$ is $\sigma$-compact
- $T$ is weakly locally compact.
From Sigma-Compact Space is Lindelöf it follows directly that:
- $T$ is Lindelöf
- $T$ is weakly locally compact.
$\Box$
2 implies 1
Now let $T = \struct {S, \tau}$ be weakly locally compact and Lindelöf.
By definition:
- $T$ is weakly locally compact if and only if every point of $S$ is contained in a compact neighborhood.
- $T$ is Lindelöf if and only if every open cover of $S$ has a countable subcover.
Thus the interiors of the compact neighborhoods are an open cover of $S$.
As $T$ is Lindelöf, this cover has a countable subcover.
Thus $T$ is the union of countably many compact subspaces.
That is, $T$ is $\sigma$-compact, and so weakly $\sigma$-locally compact.
$\blacksquare$
Also see
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Localized Compactness Properties