Where does compactness collide?

In topology, a space is compact when every open cover of it has a finite subcover, a finiteness property that generalises the closed-and-bounded subsets of Euclidean space and underwrites results like the extreme value theorem. Fréchet's 1906 Sur quelques points du calcul fonctionnel introduced the term "compact" for an early form of this idea, and Munkres's Topology gives the standard modern treatment. The referent is a structural property of a topological space.[1][2]

Unit

Whether every open cover of the space admits a finite subcover

Recovery means

Not applicable

Measured by

Established via the finite-subcover definition, or equivalents such as sequential compactness in metric spaces

In model theory, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. Chang and Keisler's Model Theory develops it as a cornerstone of the field; it descends from Gödel's completeness theorem and Mal'cev's extension to arbitrary cardinalities. The referent is a theorem about the satisfiability of theories; its "finiteness" echoes the topological property, and the two are formally linked through the topology on spaces of types, but it is a different construct about logical systems.[3]

Unit

Whether a theory has a model given that each of its finite subsets does

Recovery means

Not applicable

Measured by

Proved as a metatheorem, via the completeness theorem, ultraproducts, or the topology on the space of types