A widely connected topological space made from diamond

TL;DR

This paper constructs a unique infinite topological space that is connected yet becomes totally disconnected upon removal of any nonempty open set, using Jensen's diamond principle.

Contribution

It introduces a new topological space with unusual properties, constructed via Jensen's diamond principle, expanding understanding of connectedness and separation in topology.

Findings

Space is regular, separable, and connected.

Removing any nonempty open set yields a totally disconnected space.

Space satisfies strong Choquet property.

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 $\diamondsuit$.