Pairwise Stone space

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In mathematics and particularly in topology, a pairwise Stone space is a bitopological space ( X , τ 1 , τ 2 ) {\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})} {\displaystyle \scriptstyle (X,\tau _{1},\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:[1] If ( X , τ ) {\displaystyle \scriptstyle (X,\tau )} {\displaystyle \scriptstyle (X,\tau )} is a spectral space, then ( X , τ , τ ∗ ) {\displaystyle \scriptstyle (X,\tau ,\tau ^{*})} {\displaystyle \scriptstyle (X,\tau ,\tau ^{*})} is a pairwise Stone space, where τ ∗ {\displaystyle \scriptstyle \tau ^{*}} {\displaystyle \scriptstyle \tau ^{*}} is the de Groot dual topology of τ {\displaystyle \scriptstyle \tau } {\displaystyle \scriptstyle \tau } . Conversely, if ( X , τ 1 , τ 2 ) {\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})} {\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})} is a pairwise Stone space, then both ( X , τ 1 ) {\displaystyle \scriptstyle (X,\tau _{1})} {\displaystyle \scriptstyle (X,\tau _{1})} and ( X , τ 2 ) {\displaystyle \scriptstyle (X,\tau _{2})} {\displaystyle \scriptstyle (X,\tau _{2})} are spectral spaces.

See also

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.