Domain representable Lindelöf spaces are cofinally Polish

We prove that, for any cofinally Polish spaceX, every locally finite family of non-empty open subsets ofXis countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class ofσ-compact spaces. It turns out that, for a topological groupGwhose space has the Lindelöf Σ-property, the spaceGis domain representable if and only if it is Čech-complete. Our results solve several published open questions.