Zero-dimensionality versus connected-singleton condition in the characterization of Stone bitopological spaces

Determine whether, in the characterization of Stone bitopological spaces, the zero-dimensionality assumption can be replaced by the requirement that every connected subset (in the sense of Pervin) is a singleton; specifically, ascertain whether every bitopological space that is T0, compact, and has only singleton connected sets is equivalent to being compact and totally order-separated as in Proposition 3.1.

Background

Proposition 3.1 establishes that a bitopological space is a Stone bitopological space if and only if it is compact and zero-dimensional with T0, equivalently compact and totally order-separated.

The authors note that in a T0 and zero-dimensional bitopological space, every connected set (in Pervin’s sense) is a singleton, raising the question of whether the stronger zero-dimensionality requirement can be weakened to the more general condition that connected sets are singletons while preserving the equivalence.