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.Open setCompared with: Openness and closedness are complementary properties, but a set can have both or neither.Extreme value theoremRelated: Closedness is one part of the Euclidean characterization of compact sets.Hausdorff spaceRelated: Every singleton is closed in a Hausdorff space.Cantor setNarrower topic: The construction leaves a closed set because the removed open intervals have an open union.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.Basis (topology)Compared with: Bases generate open sets directly; closed sets follow by taking complements.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.Cantor's intersection theoremNarrower topic: Closedness is required for the nested sets in both standard forms.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.
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.Open setCompared with: Openness and closedness are complementary properties, but a set can have both or neither.Extreme value theoremRelated: Closedness is one part of the Euclidean characterization of compact sets.Hausdorff spaceRelated: Every singleton is closed in a Hausdorff space.Cantor setNarrower topic: The construction leaves a closed set because the removed open intervals have an open union.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.Basis (topology)Compared with: Bases generate open sets directly; closed sets follow by taking complements.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.Cantor's intersection theoremNarrower topic: Closedness is required for the nested sets in both standard forms.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.