Proposition 5.5.7.label Let $\seq{X_n}$ be separable topological spaces, then $\prod_{n \in \natp}X_{n}$ is separable.
Proof. For each $n \in \nat$, let $A_{n} \subset X_{n}$ be a countable dense subset. For each $N \in \nat$, let $B_{N} \subset X$ such that:
- (1)
$B_{N}$ is countable.
- (2)
For each $\bracsn{a_n}_{1}^{N} \subset \prod_{n = 1}^{N} A_{n}$, there exists $b \in B_{N}$ such that $a_{n} = b_{n}$ for all $1 \le j \le N$.
Then, $\bigcup_{N \in \natp}B_{N}$ is a countable dense subset of $\prod_{n \in \natp}X_{n}$.$\square$
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