A closed set in a topological space (X, τ) is the complement of an open set: C is closed if and only if X \ C is open. Equivalently, a set is closed if it contains all its limit points — every convergent sequence (or net) with terms in C has its limit in C.
Closed sets are closed under arbitrary intersections and finite unions, dual to the open-set axioms. The closure of a set A — the smallest closed set containing A — is obtained by adding all limit points. The closure operator cl satisfies cl(∅) = ∅, A ⊆ cl(A), cl(cl(A)) = cl(A), and cl(A ∪ B) = cl(A) ∪ cl(B) (the Kuratowski axioms), and any operator satisfying these axioms determines a topology.
In a Heyting algebra, the analogue of closure is the modality j: an element a is j-closed (or j-stable) if j(a) = a. The j-stable elements form a complete Heyting subalgebra, analogous to how the closed sets of a topological space carry their own induced topology. Topological closure is the prototypical example of the closure operators that appear throughout the semiotic universe — the semantic, syntactic, and fusion closures each stabilize different aspects of semiotic structure, just as topological closure stabilizes a set by absorbing its boundary.