Second-Countable Space is First-Countable/Proof 2

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is second-countable.

Then $T$ is also first-countable.

Proof

By definition of second-countable space, there exists a countable analytic basis $\BB \subseteq \tau$.

Then each $x \in S$ has a countable local basis:

$\BB_x := \set {U \in \BB: x \in U}$

$\blacksquare$

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