When are singletons $G_\delta$? In which topological and metric spaces are the sigletons $G_δ$,  and why?
One example is $(\mathbb{R},|\cdot|)$,  or the Euclidean multidimensional metric space.
Thank you.

Singletons are $G_\delta$'s in a  first countable $T_1$ space.  That is, for any $x$ there is a sequence $B_n$ of neighbourhoods of $x$ such that every neighbourhood of $x$ contains some $B_n$.  For any $y \ne x$, since the complement of $\{y\}$ is a neighbourhood of $x$ there is some $B_n$ that doesn't contain $y$, and so $\cap_n B_n = \{x\}$.  
An example of a space that doesn't have this property is the cofinite topology on an uncountable set.
EDIT:
A space in which singletons are $G_\delta$'s must be a $T_1$ space, but need not be first countable.  A counterexample is the quotient space $\mathbb R/\mathbb N$ (i.e. take the reals and identify all natural numbers as the same point).
