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 concept of a Stone space and its basic properties were discovered and studied by M.H. Stone (1934–1937, see [1]).

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.

References

[1]	R. Sikorski, "Boolean algebras" , Springer (1969)

Comments

An ultrafilter on a Boolean algebra is a maximal filter on the underlying ordered set.

A good up-to-date reference on Boolean algebras is [a2].

The term "Stone space" is often used as a synonym for "totally-disconnected compact space" , although others use "Boolean space" for this concept. A topological space is a Stone space if and only if it is compact, Hausdorff and totally disconnected; compact and totally separated; compact, $T_0$ and zero-dimensional; or Hausdorff and coherent.

The correspondence between Boolean algebras and their Stone spaces is a categorical duality (see Dual category); thus, if $\mathcal{B}$ and $\mathcal{B}_1$ are Boolean algebras with Stone spaces $X$ and $X_1$ respectively, the Boolean homomorphisms $\mathcal{B} \rightarrow \mathcal{B}_1$ correspond bijectively to continuous functions $X_1 \rightarrow X$. Using this, one may translate algebraic theorems about Boolean algebras into topological theorems about Stone spaces, or vice versa: for example, Sikorski's theorem that complete Boolean algebras are injective in the category of all Boolean algebras (see Injective object) corresponds to Gleason's theorem that extremally-disconnected spaces are projective in the category of Stone spaces. For more details see [a1].

References

[a1]	P.T. Johnstone, "Stone spaces" , Cambridge Univ. Press (1982) ISBN 0-521-33779-8
[a2]	S. Koppelberg, "General theory of Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 3 , North-Holland (1989)
[a3]	J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 1–3 , North-Holland (1989)