Stone space and clopen algebra

A Stone space is a topological space that is compact, Hausdorff, and has a basis of clopen subsets. (Equivalently, in the literature, a compact Hausdorff space that is totally disconnected; this library uses the clopen-basis form and does not import the connected-component characterisation.)

For a Stone space $X$ let

$\mathrm{Clop}(X):=\{C\subseteq X:C\text{ is clopen}\},$

with the Boolean operations $C\wedge C^{\mathrm{\prime }}:=C\cap C^{\mathrm{\prime }}$, $C\vee C^{\mathrm{\prime }}:=C\cup C^{\mathrm{\prime }}$, $\mathrm{\neg }C:=X\setminus C$, $0:=\mathrm{\varnothing }$, $1:=X$. These operations make $\mathrm{Clop}(X)$ a Boolean algebra (Boolean algebra and Boolean ultrafilter): the distributive and complement laws are the set-theoretic identities, and the finite unions and intersections of clopen sets are clopen by the definition of a topology.

For a Boolean algebra $B$ let

$\mathrm{Ult}(B):=\{U\subseteq B:U\text{ is a Boolean ultrafilter}\}$

be its set of ultrafilters. The ultrafilter space $\mathrm{Ult}(B)$ carries the topology generated by the sets

$[b]:=\{U\in \mathrm{Ult}(B):b\in U\}(b\in B),$

that is, the coarsest topology in which every $[b]$ is open. Compactness and Hausdorffness are not part of this definition. Under the Axiom of Choice, the ultrafilter dichotomy first shows that each $[b]$ is clopen with complement $[\mathrm{\neg }b]$, and Stone representation for Boolean algebras then proves that the resulting space is compact and Hausdorff.

• The trivial algebra. If $B$ is the one-element Boolean algebra then it has no proper filters, so $\mathrm{Ult}(B)=\mathrm{\varnothing }$; the empty space is compact, Hausdorff and has the empty basis, so it is a Stone space, and $\mathrm{Clop}(\mathrm{\varnothing })=\{\mathrm{\varnothing }\}$ is the trivial algebra. This is the convention under which the duality is total.
• Clopen basis versus total disconnectedness. Every clopen-basis compact Hausdorff space is totally disconnected, and the converse holds for compact Hausdorff spaces; the equivalence is standard but is not needed below, since only the basis property is used.
• Ultrafilter spaces are the prototypical Stone spaces under Choice. Under the Axiom of Choice, the representation theorem Stone representation for Boolean algebras and the duality Stone duality show that every Stone space is homeomorphic to some $\mathrm{Ult}(B)$ and every Boolean algebra to some $\mathrm{Clop}(X)$.