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

See also

• Bitopological space
• Duality theory for distributive lattices

Notes

1. G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, 20.

• Topology