Exercise 30.5

Proof of $(a)$. Let $X$ be a metrizable space and $A$ a countable dense subset. We claim that the set of open balls in $X$ below is a basis for $X$:

Note $B$ is countable since is in bijection with $A\times \mathbb{N}$. So let $x$ be contained in an open subset $U\subseteq X$; since $X$ is metrizable, $x\in B(x,𝜖)\subseteq U$ for some small $𝜖$. Let $n$ be such that $1∕n<𝜖∕2$. Then, since $A$ is dense, some $a\in A$ is contained in $B(x,1∕n)$, and conversely $x\in B(a,1∕n)$. By the triangle inequality, $x\in B(a,1∕n)\subseteq U$, so by Lemma $13.2$ we are done. □

Proof of $(b)$. Let $X$ be a metrizable space. Then, the set of open balls

is an open cover of $X$ for each $n\in \mathbb{N}$; since $X$ is Lindelöf, it has a countable subcover $B_n$. We claim $B{\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n\in \mathbb{N}}{B}_n$ is a basis for $X$; note it is countable since it is a countable union of countable sets. So let $x\in B(x,𝜖)\subseteq U$ as before, and let $n$ such that $1∕n<𝜖∕2$. Then, there is some $x^{\prime }\in X$ such that $B(x^{\prime },1∕n)\in B_n$ contains $x$. By the triangle inequality, $x\in B(x^{\prime },1∕n)\subseteq U$, so by Lemma $13.2$ we are done. □