Pairwise Stone space

In mathematics and particularly in topology, pairwise Stone space is a bitopological space $$\scriptstyle (X,\tau_1,\tau_2)$$ which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional.

Pairwise Stone spaces are a bitopological version of the Stone spaces.

Pairwise Stone spaces are closely related to spectral spaces.

Theorem: If $$\scriptstyle (X,\tau)$$ is a spectral space, then $$\scriptstyle (X,\tau,\tau^*)$$  is a pairwise Stone space, where $$\scriptstyle \tau^*$$  is the de Groot dual topology of $$\scriptstyle \tau$$. Conversely, if $$\scriptstyle (X,\tau_1,\tau_2)$$ is a pairwise Stone space, then both $$\scriptstyle (X,\tau_1)$$  and $$\scriptstyle (X,\tau_2)$$  are spectral spaces.