KnowraHausdorff spaceLinked fromLinked fromThe 17 pages that link to Hausdorff space, each with the reason it gives.All 17Broader topic 3Related 9Compared with 5Compact spaceRelated: Compact subsets of Hausdorff spaces are closed, a useful consequence requiring this extra axiom.Topological spaceBroader topic: The Hausdorff condition strengthens how the topology separates distinct points.HomeomorphismRelated: The Hausdorff condition makes compact-to-Hausdorff continuous bijections automatically homeomorphisms.Stone–Čech compactificationRelated: The compactification and its universal target spaces are required to be Hausdorff.Convergent sequenceRelated: The Hausdorff separation property ensures that convergent sequences have unique limits.Sequential compactnessRelated: Hausdorffness ensures sequence limits are unique, though it is not required by the definition.Normal spaceCompared with: Hausdorff separation concerns points, while normality separates arbitrary disjoint closed sets.Sierpiński spaceCompared with: Its two points cannot be separated by disjoint open neighborhoods.Felix HausdorffBroader topic: This separation property is the condition most directly associated with Hausdorff’s name.Separation axiomBroader topic: The Hausdorff condition gives a direct, pairwise separation of distinct points.Urysohn's lemmaCompared with: Hausdorff separation concerns points, while the lemma requires normality for closed sets.Tietze extension theoremCompared with: Hausdorff separation alone is weaker than the normality required by the theorem.Urysohn's metrization theoremRelated: The theorem’s regularity hypothesis entails point separation in the standard T1 setting.Tube lemmaCompared with: The tube lemma does not require the Hausdorff separation condition.Pavel AlexandrovRelated: The one-point compactification has its standard form for locally compact Hausdorff spaces.Cantor's intersection theoremRelated: Hausdorffness helps distinguish compact-set formulations from stronger uniqueness claims.Stone's representation theorem for Boolean algebrasRelated: The theorem's representing space separates distinct ultrafilters by neighborhoods.
KnowraHausdorff spaceLinked fromLinked fromThe 17 pages that link to Hausdorff space, each with the reason it gives.All 17Broader topic 3Related 9Compared with 5Compact spaceRelated: Compact subsets of Hausdorff spaces are closed, a useful consequence requiring this extra axiom.Topological spaceBroader topic: The Hausdorff condition strengthens how the topology separates distinct points.HomeomorphismRelated: The Hausdorff condition makes compact-to-Hausdorff continuous bijections automatically homeomorphisms.Stone–Čech compactificationRelated: The compactification and its universal target spaces are required to be Hausdorff.Convergent sequenceRelated: The Hausdorff separation property ensures that convergent sequences have unique limits.Sequential compactnessRelated: Hausdorffness ensures sequence limits are unique, though it is not required by the definition.Normal spaceCompared with: Hausdorff separation concerns points, while normality separates arbitrary disjoint closed sets.Sierpiński spaceCompared with: Its two points cannot be separated by disjoint open neighborhoods.Felix HausdorffBroader topic: This separation property is the condition most directly associated with Hausdorff’s name.Separation axiomBroader topic: The Hausdorff condition gives a direct, pairwise separation of distinct points.Urysohn's lemmaCompared with: Hausdorff separation concerns points, while the lemma requires normality for closed sets.Tietze extension theoremCompared with: Hausdorff separation alone is weaker than the normality required by the theorem.Urysohn's metrization theoremRelated: The theorem’s regularity hypothesis entails point separation in the standard T1 setting.Tube lemmaCompared with: The tube lemma does not require the Hausdorff separation condition.Pavel AlexandrovRelated: The one-point compactification has its standard form for locally compact Hausdorff spaces.Cantor's intersection theoremRelated: Hausdorffness helps distinguish compact-set formulations from stronger uniqueness claims.Stone's representation theorem for Boolean algebrasRelated: The theorem's representing space separates distinct ultrafilters by neighborhoods.