Pairwise Stone space

In mathematics and particularly in topology, a pairwise Stone space is a bitopological space ${\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})}$ that is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional.

Pairwise Stone spaces are a bitopological version of Stone spaces.

Pairwise Stone spaces are closely related to spectral spaces.

Theorem:¹ If ${\displaystyle \scriptstyle (X,\tau )}$ is a spectral space, then ${\displaystyle \scriptstyle (X,\tau ,\tau ^{*})}$ is a pairwise Stone space, where ${\displaystyle \scriptstyle \tau ^{*}}$ is the de Groot dual topology of ${\displaystyle \scriptstyle \tau }$ . Conversely, if ${\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})}$ is a pairwise Stone space, then both ${\displaystyle \scriptstyle (X,\tau _{1})}$ and ${\displaystyle \scriptstyle (X,\tau _{2})}$ are spectral spaces.