Connected regular space

This article describes a property of topological spaces obtained as a conjunction of the following two properties: regularity and connectedness

Definition

A connected regular space is a topological space which is both connected and regular.

Facts

• Any connected regular space which has at least two points, is uncountable. The proof of this is roundabout: we first observe that any countable space is Lindelof, then the fact that any regular Lindelof space is normal, and then the fact that any connected normal space with at least two points has cardinality at least that of the reals. Note that this does not prove that any connected regular space has cardinality at least that of the reals.