closed set (topological)

If open sets are the “roomy” sets where every point has wiggle room, closed sets are their photographic negatives: a set is closed precisely when its complement is open. Intuitively a closed set contains all of its own boundary — it includes every point it gets arbitrarily close to and never leaks any limit out.

Formally, in a topological space (X, T) a subset C is closed if and only if X minus C belongs to T. By taking complements of the open set axioms (and De Morgan's laws), the closed sets satisfy dual rules: X and the empty set are closed; any finite union of closed sets is closed; and any intersection (even infinite) of closed sets is closed.

Closedness can be re-expressed in terms of limits, and this is the version analysts use most: a set C is closed if and only if it contains the limit of every convergent sequence of its points (in spaces where sequences suffice to detect closure, such as metric spaces). This is why “closed” and “contains its limits” are nearly synonymous in elementary analysis — but in general topological spaces one must use nets or filters rather than sequences for the full statement.