G-Delta Sets and First Countability (3.30.1)

James Munkres Topology, chapter 3.30, exercise 1:

(a) A $G_δ$ set in a space $X$ is a countable intersection of open sets of $X$. Show that in a first-countable $T_1$ space, every one-point set is a $G_δ$ set.
(b) There is a familiar space wherein every one-point set is a $G_δ$ set, which is nevertheless not first countable. What is it?

Proof: (a) Let $x∈X$, and let $\{B_i\}$ be a countable basis at $x$. Suppose $y∈∩B_i$ while $y \neq x$. Then since $X$ is $T_1$, let $U$ be a neighborhood of $x$ not containing $y$. We see $U$ is open and contains $x$, yet contains no element of the basis $\{B_i\}$ seeing as $y \not \in U$, a contradiction.

(b) Let $ℝ^\omega$ be under the box topology. Then given $x∈ℝ^\omega$, when we let $U_n=∏(x_n-1/n,x_n+1/n)$ we see $∩U_n=\{x\}$, so that every one-point set in $ℝ^\omega$ is $G_δ$. To show that $ℝ^\omega$ is not first countable, suppose $\{B_n\}$ is a countable basis at any particular point $x$. Then for each $n∈ℕ$, we may choose an interval $(a_n,b_n)$ such that $x_n∈(a_n,b_n) \subset π_n(B_n)$. Hence, let $U=∏(a_n,b_n)$; we see $x∈U$ yet $B_n \not \subseteq U$ for all $n$ since $π_n(B_n) \not \subseteq π_n(U)$, so that $\{B_n\}$ is not a countable basis at $x$, a contradiction.$~\square$