Let $X = \{(x,\alpha):x \in A_\alpha, \alpha \in \omega_1\}$, in which $\{A_\alpha: \alpha \in \omega_1\}$ are pairwise disjoint, countable dense subsets of the (open) unit interval $(0,1)$. The topology of the space $X$ is generated by the sets $O(a,b,\alpha)=\{(x,\beta) \in X: a < x < b, \alpha \le \beta < \omega_1\}, 0 \le a < b \le 1, \alpha \in \omega_1\;.$ $X$ is Hausdorff (not regular), Lindelöf, first countable, CCC, but not separable (These have been proved). Is $X$ submetrizable?
Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.