Second-Countable Space is Lindelöf

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
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