Second-Countable Space is Lindelöf
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Theorem
Let $T = \struct {S, \tau}$ be a topological space which is second-countable.
Then $T$ is also a Lindelöf space.
Proof
Let $T$ be second-countable.
Then by definition its topology has a countable basis.
Let $\BB$ be this countable basis.
Let $\CC$ be an open cover of $T$.
Every set in $\CC$ is the union of a subset of $\BB$.
So $\CC$ itself is the union of a subset of $\BB$.
This union of a subset of $\BB$ is therefore a countable subcover of $\CC$.
That is, $T$ is by definition Lindelöf.
$\blacksquare$
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: Countability Axioms and Separability