Compactness

Defintion

Let $X$ be a topological space. Say that $X$ is compact iff every open cover has a finite subcover.

Equivalently, $X$ is compact iff every family of closed subsets with the finite intersection property has non-empty intersection.

That is, if $(C_i{)}_{i\in I}$ is a family of closed subsets of $X$ such that every finite intersection

$$C_{i_1}\cap \cdots \cap C_{i_n}\ne \varnothing ⟹\bigcap _{i\in I}{C}_i\ne \varnothing$$

Other appearances

Compactness appears in many areas of mathematics. The details differ, but the common theme is usually that some infinite object is controlled by finite approximations.

Stone Spaces

The connection between topological compactness and logical compactness becomes especially concrete through Stone spaces.

Metric spaces

In metric spaces, compactness is closely related to sequential compactness: every sequence has a convergent subsequence. In spaces like ${\mathbb{R}}^n$, the Heine-Borel Theorem says that a subset is compact iff it is closed and bounded.

This gives compactness the flavour of being “finite in extent”, even when the space has infinitely many points.

Algebra and logic

In logic, compactness says that satisfiability of an infinite theory is determined by satisfiability of its finite fragments.

This can also be viewed topologically using Stone spaces, where theories correspond to intersections of clopen sets.

In logic, the compactness of a system says that if a sentence $\phi$ follows from a possibly infinite set of premises $\Gamma$, then it already follows from some finite subset ${\Gamma }_0\subseteq \Gamma$:

$$\Gamma \models \phi ⟹\exists {\Gamma }_0{\subseteq }_{\mathrm{fin}}\Gamma ,{\Gamma }_0\models \phi .$$

Equivalently, a theory $T$ is satisfiable iff every finite subset of $T$ is satisfiable.

This is closely analogous to the closed-set formulation of topological compactness. Each sentence $\phi$ as defining the class of models satisfying it:

$$[\phi ]=\{M:M\models \phi \}.$$

A theory $T$ then defines an intersection:

$$\bigcap _{\phi \in T}[\phi ].$$

The theory is satisfiable exactly when this intersection is non-empty. Thus logical compactness says:

If every finite collection of constraints has a simultaneous solution, then all the constraints have a simultaneous solution.

This is the same formal shape as topological compactness in its finite-intersection form.

Category theory

In category theory, an object $C$ is often called compact, or finitely presentable, when maps out of $C$ commute with filtered colimits:

$$\mathrm{Hom}(C,{\mathrm{colim}}_i{X}_i)\cong {\mathrm{colim}}_i\mathrm{Hom}(C,X_i).$$

Informally, this says that any map from $C$ into a filtered colimit already factors through some finite stage. Again, the theme is that a map out of a compact object cannot see genuinely infinite construction all at once.