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:^[1] 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.

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.