Can any result on metric spaces be applied to metrizable spaces?

Question

Can you give me a result (theorem, lemma, proposition) that doesn't hold on a metrizable space, but does on a metric space? This question is a bit vague, because you could easily say "A metric space is [insert metric space definition], a metrizable space is not a metric space", and so we have an example. I don't know how to classify the kind of result I am looking for, however it shouldn't be exploitative like my example, but actually meaningful.

Answer 1

As you've pointed out, there's a trivial negative answer if we interpret the question too broadly. So we need to restrict attention to "nice" results/properties of spaces.

In my opinion the most natural thing to do is to look at properties phrased in the language of topology alone (so "is a metric space" doesn't count) and interpret a statement like "every metric space has property $P$" as "for every metric space $(X,d)$ the induced topological space has property $P$."

This then gives a trivial positive answer: metrizable spaces being those induced by metric spaces, this immediately implies that every metrizable space has property $P$.

It's not obvious to me how to refine the question to avoid these two trivialities at once, and in fact I think we shouldn't. I think the second triviality is actually the answer to the question. Topology cannot distinguish metrizable spaces from specific metric spaces, and indeed that's sort of the point of metrizability in the first place: it's a topological property which captures exactly the topological consequences of being a metric space.

Answer 2

If the theorem expresses a property of metric spaces, and if that property does not actually involve the metric in its statement, then that theorem holds for a metrizable space. Here's an example:

Theorem: Every metric space is first countable.

Since first countability does not involve the actual metric in its formulation, we could just as well state this theorem in the form "Every metrizable space is first countable."

But some theorems about metric spaces put conditions on the metric, and draw conclusions about metrics which satisfy that condition. In that case the theorem need not hold for general metrizable spaces, even if the conclusion does not mention the metric. Here's an example:

Baire Category Theorem: If $X$ is a complete metric space then for every countable collection $\{U_i \mid i=1,2,\ldots\}$ of dense open sets, the intersection $\cap_{i=1}^\infty U_i$ is dense.