Exercise 2.24

We can assume $X$ is infinite. Pick $\delta >0$, pick an $x_1\in X$, and inductively pick $x_{n+1}$ such that $d(x_i,x_{n+1})>\delta$ for all $1\le i<n+1$ if possible. This process must terminate. Otherwise, ${\{x_i\}}_{i=1}^{\infty }$ would have a limit point $x$, and $N_{\frac{\delta }{2}}(x)$ would contain two distinct points of the sequence, a contradiction.

$X$ can therefore be covered with a finite number of neighborhoods of radius $\delta$ around a finite number of points $x_1,\cdots ,x_k$. Let $B$ be the collection of all such neighborhoods for $\delta =\frac{1}{n}$, $n\in \mathbb{N}$. This is a countable collection. Given a neighborhood $N_{𝜖}(x)$ of $x\in X$, pick $k$ such that $\frac{1}{k}<\frac{𝜖}{2}$. Then $x$ is covered by some neighborhood $N\in B$ of radius $\frac{1}{k}$, and $N\subset N_{𝜖}(x)$. This shows that $B$ is a countable base.