pairwise Stone space

{{one source|date=January 2023}}

In mathematics and particularly in topology, a pairwise Stone space is a bitopological space $\backslash scriptstyle\; (X,\backslash tau\_1,\backslash tau\_2)$ 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:G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, 20. If $\backslash scriptstyle\; (X,\backslash tau)$ is a spectral space, then $\backslash scriptstyle\; (X,\backslash tau,\backslash tau^*)$ is a pairwise Stone space, where $\backslash scriptstyle\; \backslash tau^*$ is the de Groot dual topology of $\backslash scriptstyle\; \backslash tau$ . Conversely, if $\backslash scriptstyle\; (X,\backslash tau\_1,\backslash tau\_2)$ is a pairwise Stone space, then both $\backslash scriptstyle\; (X,\backslash tau\_1)$ and $\backslash scriptstyle\; (X,\backslash tau\_2)$ are spectral spaces.