KnowraClosed setLinked fromLinked fromThe 17 pages that link to Closed set, each with the reason it gives.All 17Related 13Narrower topic 2Compared with 2Compact spaceRelated: Closedness combines with boundedness to characterize compact subsets of Euclidean space, but not generally.Extreme value theoremRelated: Closedness is one part of the Euclidean characterization of compact sets.Hausdorff spaceRelated: Every singleton is closed in a Hausdorff space.Heine–Borel theoremRelated: Closedness is one of the two conditions that characterize compact subsets of Euclidean space.Convergent sequenceRelated: In metric spaces, a set is closed exactly when it contains limits of its convergent sequences.Normal spaceRelated: Normality tests pairs of disjoint sets of this kind.Portmanteau theoremRelated: The theorem bounds the limiting upper probability of a closed set by its limit-measure probability.Separation axiomRelated: Several separation conditions ask whether closed sets can be separated from points or other closed sets.Urysohn's lemmaRelated: The two sets separated by the lemma must be closed.Tietze extension theoremRelated: The function is initially defined on a closed subset, a key condition of the theorem.Hopf–Rinow theoremRelated: The theorem guarantees compactness for closed bounded subsets, not arbitrary bounded ones.Closed-subgroup theoremRelated: The subgroup hypothesis is topological closure in the ambient Lie group.Kuratowski's closure-complement problemRelated: Applying closure produces a closed set, one of the basic stopping points.