A star is $\text{St}(A, \mathcal{U}) = \bigcup\{U\in \mathcal{U} : U\cap A\neq\emptyset\}$. A development is a sequence of open covers, $(\mathcal{U}_n)_{n\in\mathbb{N}}$ such that $\text{St}(x, \mathcal{U}_n)$ forms neighbourhood basis for all $x\in X$. We call $\mathcal{V}$ a star-refinement of $\mathcal{U}$ if $\{\text{St}(V, \mathcal{V}) : V\in\mathcal{V}\}$ is a refinement of $\mathcal{U}$.
Theorem. (Alexandroff-Urysohn) A Hausdorff space $X$ is metrizable iff there exists a development $(\mathcal{U}_n)_{n\in\mathbb{N}}$ of $X$ such that $\mathcal{U}_{n+1}$ is a star-refinement of $\mathcal{U}_n$ for all $n\in\mathbb{N}$.
What are some applications of this theorem (to topology and mathematics in general)? I never seen any, hence the question.
One application that I could find, is this result:
If $X$ is metrizable and $f:X\to Y$ is perfect, then $f[X]$ is metrizable.
I think this might be as far as applications go without going into applications to other metrization theorems and to generalized metric spaces.