CONSTRUCTING REGULAR SPACES THAT ARE NOT COMPLETELY REGULAR

Constructions of regular spaces on which every continuous real-valued function is constant effectively rely on the condensation of twins in a regular, not completely regular, space. It is known that from any regular non-normal space one can contruct a space of the latter type. The main result presented here produces first countable spaces of the former type from any regular first countable space that has twins. This construction preserves properties of the starting space, such as developability and countable paracompactness. As a consequence, it is consistent that there exists a separable countably paracompact Moore space on which every continuous real-valued function is constant.