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

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