Is there something wrong with this statement?

Question

Is this sentence OK or is there something wrong?

"Hausdorff is a very weak separation axiom for one to discuss whether the space is metrizable."

Do you have a better expression?

Answer 1

Your sentence is perfectly understandable, but it does sound a bit odd to a native speaker of English. Does this convey more or less what you want to say?

Metrizable spaces satisfy very strong separation axioms; the assumption that a space is Hausdorff is too weak to tell us anything very useful about its possible metrizability.

Answer 2

Well, Every topology induced by a metric is going to be Hausdorff, since metrics are nondegenerate. There are lots of so-called metritrization theorems. Unfortunately Hausdorff by itself is not good enough for having a metric.

A better expression might be: While the condition that a topology be Hausdorf is necessary for it to be metrizable, it is too weak to give us that such a space is metrizable.

To find an example, you might try to find a Hausdorff topology that is not first countable.