How can the implication metrizable -> first countable be generalized?

Question

I'm making a contribution to the pi-Base to automatically deduce certain spaces are not metrizable based on the lack of first-countability. How can this theorem be generalized to deduce more [non-]properties?

Answer 1

The pi-Base actually already knows pseudometrizable implies first-countable. It could also be improved by adding local metrizability as a sufficient condition, but pseduometrizable doesn't imply local metrizability, so there's a bit of a disconnect there.

So here's an approach. Say a space is locally [pseudo]metrizable if every point has a [pseudo]metrizable neighborhood (and therefore a basis of such neighborhoods, so it's not a "weak" local property).

Then we have the following:

• Metrizable ->(1) pseudometrizable ->(2) locally pseudometrizable ->(3) first countable
• Metrizable ->(2) locally metrizable ->(1) locally pseudometrizable ->(3) first countable

Proofs:

1. Every metric is a pseduometric.
2. The entire space is the desired neighborhood.
3. Take a pseduometrizable neighborhood with pseudometric $\rho$. Then $\{B_\rho(x,\epsilon):\epsilon\in\mathbb Q^+\}$ is a countable local base at $x$.

(I'm unaware of any actual use of "locally pseduometrizable" in the literature, but it seems to be a natural intermediate concept.)