A topological space X is compact if every open cover has a finite subcover: whenever X = ⋃ᵢ Uᵢ for a collection of open sets Uᵢ, some finite subcollection already covers X. Compactness is a finiteness condition — it ensures that the space cannot “escape to infinity” and that local information can always be assembled from finitely many pieces.
In ℝⁿ, compactness is equivalent to being closed and bounded (the Heine-Borel theorem). But the topological definition works in any space: the unit interval [0,1] is compact, ℝ is not, and the Cantor set is compact despite being uncountable and nowhere dense.
Compactness has far-reaching consequences. Continuous real-valued functions on a compact space attain their maximum and minimum. A continuous map from a compact space to a Hausdorff space is automatically a closed map. The product of any collection of compact spaces is compact (Tychonoff’s theorem), a result whose proof requires the axiom of choice and whose consequences pervade analysis, algebra, and logic.
In the semiotic universe, fragments are the compact elements of the Heyting algebra H: a fragment F is a finitely generated sub-Heyting-algebra that can be covered by finitely many generators. This compactness ensures that reasoning within a fragment is finitarily manageable — it is the formal basis for the claim that fragments support scoped, local reasoning rather than requiring the entire algebra at once.