Cosmicity of Cometrizable Spaces

A space X is cometrizable if X has a coarser metric topology such that each point of X has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space X : (a) X is the continuous image of a separable metric space; (b) Xw is hereditarily separable and hereditarily Lindelöf, (c) X2 has no uncountable discrete subspaces; (d) X is a Lindelöf semimetric space; (e) X has the pointed ccc. This result is a corollary to our main result which states that, under PFA, if Ais a cometrizable space with no uncountable discrete subspaces, then either X is the continuous image of a separable metric space or X contains a copy of an uncountable subspace of the Sorgenfrey line.