Second countability axiom

Second countable implies Lindelöf

Let be a topological space. If is second-countable then it is Lindelöf.

Proof

Let be a countable topological basis of , and be an open cover. Let such that

And for every let such that . Since every is the union of some family of with , is a countable open cover of and therefore is too.

The proof relies on the Axiom of Choice.

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