Exercise 2.23

Let $B$ be a countable, dense subset of $X$. Note that the collection $B=\{N_{\frac{1}{n}}(x):x\in B,n\in \mathbb{N}\}$ is countable. We claim that it is a base. Let $x\in U$, $U$ open. Then $N_{𝜖}(x)\subset U$ for some $𝜖$. Pick $k$ such that $\frac{1}{k}<\frac{𝜖}{2}$. By density, we can find $y\in B$ such that $d(x,y)<\frac{1}{k}$. Then $x\in N_{\frac{1}{k}}(y)$, which is in $B$, and $N_{\frac{1}{k}}(y)\subset N_{𝜖}(x)\subset U$. This finishes the proof.