• Source: Pairwise Stone space

In mathematics and particularly in topology, pairwise Stone space is a bitopological space




(
X
,

τ

1


,

τ

2


)



{\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




(
X
,
τ
)



{\displaystyle \scriptstyle (X,\tau )}

is a spectral space, then




(
X
,
τ
,

τ

∗


)



{\displaystyle \scriptstyle (X,\tau ,\tau ^{*})}

is a pairwise Stone space, where





τ

∗





{\displaystyle \scriptstyle \tau ^{*}}

is the de Groot dual topology of




τ



{\displaystyle \scriptstyle \tau }

. Conversely, if




(
X
,

τ

1


,

τ

2


)



{\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})}

is a pairwise Stone space, then both




(
X
,

τ

1


)



{\displaystyle \scriptstyle (X,\tau _{1})}

and




(
X
,

τ

2


)



{\displaystyle \scriptstyle (X,\tau _{2})}

are spectral spaces.




See also


Bitopological space
Duality theory for distributive lattices




Notes

Kata Kunci Pencarian:

• Pairwise Stone space
• Spectral space
• Priestley space
• Duality theory for distributive lattices
• Hilbert space
• Stone–von Neumann theorem
• Borromean rings
• Topological property
• Polyadic space
• Long line (topology)