# Second-Countable Space is First-Countable

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## Theorem

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

Then $T$ is also first-countable.

## Proof

By definition $T$ is second-countable if and only if its topology has a countable basis.

Consider the entire set $S$ as an open set.

From Set is Open iff Neighborhood of all its Points, $S$ has that property.

As $T$ has a countable basis, then (trivially) every point in $T$ has a countable local basis.

So a second-countable space is trivially first-countable.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$: Countability Properties - 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 3$: Countability Axioms and Separability