title: Pairwise Stone space
text: # 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.
  
  