A widely connected topological space made from diamond

Abstract

We give the construction of an infinite topological space with unusual properties. The space is regular, separable, and connected, but removing any nonempty open set leaves the remainder of the space totally disconnected (in fact, totally separated). The space is also strongly Choquet (in fact, satisfies an even stronger condition) and has a basis with nice properties. The construction utilizes Jensen's diamond principle $\diamondsuit$.