# Stone space

of a Boolean algebra $\mathcal{B}$
A totally-disconnected compact space $(X,\mathfrak{T})$ whose field of all open-and-closed sets is isomorphic to $\mathcal{B}$. This space is defined canonically from $\mathcal{B}$ in the following way: $B$ is the set of all ultrafilters of $\mathcal{B}$, while the topology $\mathfrak{T}$ is generated by the family of subsets of the form $U_A = \{ \xi \in X : A \in \xi \}$, where $A$ is an arbitrary element of $\mathcal{B}$. Instead of ultrafilters, the set of maximal ideals, of two-valued homomorphisms or of two-valued measures on $\mathcal{B}$ with an appropriate topology may be used. Isomorphic Boolean algebras have homeomorphic Stone spaces. Every totally-disconnected compact space is the Stone space of the Boolean algebra of its open-and-closed sets.
The Stone space of a Boolean algebra is metrizable if and only if the Boolean algebra is countable. A Boolean algebra is complete if and only if its Stone space is extremally disconnected (i.e. if the closure of any open set in the space is open). The perfect Cantor set is the Stone space of a countable atomless Boolean algebra (they are all isomorphic). The generalized Cantor discontinuum $D^m$ is the Stone space of the free Boolean algebra with $m$ generators.
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