Pairwise Stone space
In mathematics and particularly in topology, pairwise Stone space is a bitopological space [math]\displaystyle{ \scriptstyle (X,\tau_1,\tau_2) }[/math] 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 [math]\displaystyle{ \scriptstyle (X,\tau) }[/math] is a spectral space, then [math]\displaystyle{ \scriptstyle (X,\tau,\tau^*) }[/math] is a pairwise Stone space, where [math]\displaystyle{ \scriptstyle \tau^* }[/math] is the de Groot dual topology of [math]\displaystyle{ \scriptstyle \tau }[/math] . Conversely, if [math]\displaystyle{ \scriptstyle (X,\tau_1,\tau_2) }[/math] is a pairwise Stone space, then both [math]\displaystyle{ \scriptstyle (X,\tau_1) }[/math] and [math]\displaystyle{ \scriptstyle (X,\tau_2) }[/math] are spectral spaces.
See also
Notes
- ↑ G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, 20.
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