Need an example of a space which is not first countable Give an example of a space which is NOT first countable & in which every singleton set is : $ G_\delta $ . I have just found out one example where the space is NOT first countable is: any uncountable set with co-finite topology, BUT cannot proceed any further.
 A: Take any countable dense subset $D$ of $\{0,1\}^\mathfrak{c}$ (this space is well-known to be separable). Being countable and Tychonoff already gives us that every point of $D$ is a $G_\delta$, $D$ is hereditarily Lindelöf (so has all separation properties one could wish for) and its weight is the weight of $\{0,1\}^\mathfrak{c}$, so also $\mathfrak{c}$. One can even show that this is the minimal cardinality of a local base at any of its points.
A: Let $F$ be the set of free ultrafilters on $\Bbb N$, and let $X=\Bbb N\cup F$. Each point of $\Bbb N$ is isolated. If $p\in F$, the family $$\big\{\{p\}\cup U:U\in p\big\}$$ is a local base at $p$. $X$ is not first countable at any point of $F$. For $n\in\Bbb N$ let $U_n=\{k\in\Bbb N:k>n\}$; then $U_n\in p$ for each $p\in P$, so for each $p\in P$ we have $$\{p\}=\bigcap_{n\in\Bbb N}\big(\{p\}\cup U_n\big)\;,$$ showing that $\{p\}$ is a $G_\delta$. $X$ is Hausdorff, since if $p,q\in P$ with $p\ne q$, there is an $A\subseteq\Bbb N$ such that $A\in p$ and $\Bbb N\setminus A\in q$, and $\{p\}\cup A$ and $\{q\}\cup(\Bbb N\setminus A)$ are disjoint open nbhds of $p$ and $q$. Clearly $X$ has a clopen base, so $X$ is zero-dimensional and Tikhonov.
(This space is $\kappa\Bbb N$, the Katětov extension of $\Bbb N$; it’s an $H$-closed space, meaning that it’s closed in every Hausdorff space in which it’s embedded.)
