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**ModusPonens** Munkres' Topology. Section 30, exercise 1-a says: "Show that in a first-countable $\displaystyle T_1$ space, every one-point set is a $\displaystyle G_{ \delta}$ set"

Now, I've seen the solution to this exercise and I believe it's wrong, because it assumes that a countable basis at x is a family of sets that get "arbitrarily small", that is, it's like a countable collection of balls of radius 1/n (I know the space doesn't need to be metrizable, I'm just giving the intuition). This assumption is wrong because there could be a strange topology in $\displaystyle R$ in which the neighbourhoods of 0 would be intervals of the form (-1-1/n,1+1/n). This is a countable basis at 0.

Now the counter example: $\displaystyle R$ with the finite complement topology. In this topology, it is $\displaystyle T_1$ and first countable, but no one-point set is a $\displaystyle G_{ \delta}$ set.