Stone ultrafilter space and its clopen basis

For a Boolean algebra $B$, let $\mathrm{Ult}(B)$ be its set of ultrafilters and put $[b]=\{U\in \mathrm{Ult}(B):b\in U\}$. Give it the topology consisting of all unions of such sets. The filter tests give

$[0]=\mathrm{\varnothing },[1]=\mathrm{Ult}(B),[b\wedge c]=[b]\cap [c],[\mathrm{\neg }b]=\mathrm{Ult}(B)\setminus [b],[b\vee c]=[b]\cup [c]\mathrm{.}$

These equations ensure that the proposed unions form a topology: the empty union and whole space occur, arbitrary unions are unions of basic sets, and finite intersections distribute over such unions. The sets $[b]$ form a clopen basis. This definition does not assert compactness or the existence of ultrafilters.

A Stone space is a compact, Hausdorff space with a basis of clopen subsets. The empty space is allowed. For any space $X$, write $\mathrm{Clop}(X)$ for its clopen subsets, with union, intersection, complement relative to $X$, and bounds $\mathrm{\varnothing },X$; finite unions and intersections of clopens and their complements are clopen, so these form a Boolean algebra. The trivial Boolean algebra has no ultrafilters and thus gives the empty space.