Stone Space

Given a Boolean algebra $B$, its Stone space $\mathrm{Stone}(B)$ is the space of ultrafilters on $B$. Equivalently, its points can be thought of as complete consistent choices of truth values for the propositions in $B$.

For each element $b\in B$, define

$$U_b=\{\mathcal{U}\in \mathrm{Stone}(B):b\in \mathcal{U}\}.$$

These sets are both open and closed, and they form a basis for the topology. The resulting space is compact, Hausdorff, and totally disconnected.

In the case of propositional logic, the Boolean algebra is the algebra of propositions modulo logical equivalence. A point of the Stone space is then a complete truth assignment, or more generally a complete consistent theory.

Under this interpretation:

• propositions correspond to clopen subsets;
• conjunction corresponds to intersection;
• disjunction corresponds to union;
• negation corresponds to complement;
• consistency corresponds to non-empty intersection.

So compactness of the Stone space says that if every finite subset of a set of propositions is satisfiable, then the whole set is satisfiable.

Thus, at least for propositional logic, logical compactness is literally topological compactness of an associated Stone space.