# Cofinite topology on infinite set coming from a metric?

How do I go about proving that cofinite topology on an infinite set does not come from a metric?

I was thinking about proving it on $\mathbb{N}$ first then generalize it on any infinite set. Because for any infinite set I can find a canonical bijection of $\mathbb{N}$ embedded in it and I reduce to the subspace topology on that copy of $\mathbb{N}$.

Is this thought process correct? Thank you.

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Hint: the metric topology is Hausdorff. –  JHF Apr 6 '13 at 18:46

Hint: Note that every two cofinite sets have a cofinite intersection. Therefore the topology cannot separate two points by disjoint open sets.

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Ok, what you mention is called $T_2$ right? Cofinite topology on infinite sets satisfy $T_1$ but not $T_2$ right? –  mez Apr 6 '13 at 19:28
@mezhang: Yes, $T_2$, or Hausdorff, as JHF mentioned in the comment. –  Asaf Karagila Apr 6 '13 at 19:30