Sigma-Compact Space is Lindelöf
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Theorem
Every $\sigma$-compact space is a Lindelöf space.
Proof
Let $T = \struct {S, \tau}$ be a $\sigma$-compact space.
By definition:
- $T$ is a Lindelöf space if and only if every open cover of $X$ has a countable subcover.
By definition of $\sigma$-compact space, $T = \bigcap \TT$ where $\TT$ is the union of countably many compact subspaces.
Let $\CC$ be an open cover of $T$.
Each element of $\TT$ is covered by a finite number of elements of $\CC$.
Hence $T$ is covered by a countable union of a finite number of elements of $\CC$.
Hence $\CC$ has a countable subcover.
Hence the result.
$\blacksquare$
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: Global Compactness Properties