1. ## Weakest Metrizable Topology?

Given a set $X$ and a nonmetrizable topology $\mathcal{T}$, is the intersection $\bigcap\left\{\mathcal{U}\subseteq\mathcal{P}(X)| \mathcal{U}\text{ is metrizable and }\mathcal{T}\subset\mathcal{U}\right\}$ metrizable?

If not, are there properties that $\mathcal{T}$ could have that would make that intersection metrizable?

2. ## Re: Weakest Metrizable Topology?

Originally Posted by SlipEternal
Given a set $X$ and a nonmetrizable topology $\mathcal{T}$, is the intersection $\bigcap\left\{\mathcal{U}\subseteq\mathcal{P}(X)| \mathcal{U}\text{ is metrizable and }\mathcal{T}\subset\mathcal{U}\right\}$ metrizable?

If not, are there properties that $\mathcal{T}$ could have that would make that intersection metrizable?
What have you tried? It seems like you could, if its true, prove this using Nagata-Smirnov.

3. ## Re: Weakest Metrizable Topology?

That looks like an excellent place to start. Thank you. I will get back to you with any results that I find.