Contrapositive of metric space implies Hausdorff We know that any metric space is Hausdorff with respect to the topology induced by the metric, or that any metrizable space is Hausdorff. Then the contrapositive is:
not Hausdorff $\stackrel{?}{\Rightarrow}$ not metrizable ?
I was thinking of a very weird contrapositive, which is not Hausdorff $\Rightarrow$ not metric space which does not make any sense
 A: If I understand the question correctly, it is readily resolved by terminological precision. The confusion arises because we say things like "a metric space is Hausdorff" when we actually mean that the topological space obtained by equipping the underlying set of the metric space with the topology induced by the metric of the metric space is Hausdorff. The contrapositive of that more precise statement is unproblematic: If a topological space is not Hausdorff, then it is not a topological space obtained by equipping the underlying set of a metric space with the topology induced by the metric of the metric space.
A: If you restrict yourself to just thinking about metric spaces, then since every space you consider is metric, hence Hausdorff, saying "a space which is not Hausdorff is not a metric space" is a vacuous (but true) statement.
If you instead consider the broader class of topological spaces, then there is more content to this statement, though you have to alter it slightly.  It becomes

Every topological space which is not Hausdorff is not metrizable.

And we do have examples of topological spaces which are not Hausdorff.  For example, given any (nonempty) set $X$ the family $\mathcal{O} = \{ \varnothing , X \}$ is a topology on $X$.  If $| X | > 1$, then this topology is not Hausdorff, and consequently there is no metric on $X$ compatible with this topology.
