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- Cocountable_topology abstract "The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X.Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X. It is also T1, as all singletons are closed. If X is an uncountable set, any two open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique. Since compact sets in X are finite subsets, all compact subsets are closed, another condition usually related to Hausdorff separation axiom.The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.".
- Cocountable_topology wikiPageID "801246".
- Cocountable_topology wikiPageLength "1867".
- Cocountable_topology wikiPageOutDegree "21".
- Cocountable_topology wikiPageRevisionID "612217974".
- Cocountable_topology wikiPageWikiLink Category:General_topology.
- Cocountable_topology wikiPageWikiLink Cocountability.
- Cocountable_topology wikiPageWikiLink Cocountable.
- Cocountable_topology wikiPageWikiLink Cofinite_topology.
- Cocountable_topology wikiPageWikiLink Cofiniteness.
- Cocountable_topology wikiPageWikiLink Compact_space.
- Cocountable_topology wikiPageWikiLink Complement_(set_theory).
- Cocountable_topology wikiPageWikiLink Connected_space.
- Cocountable_topology wikiPageWikiLink Countable_set.
- Cocountable_topology wikiPageWikiLink Counterexamples_in_Topology.
- Cocountable_topology wikiPageWikiLink Discrete_space.
- Cocountable_topology wikiPageWikiLink Discrete_topology.
- Cocountable_topology wikiPageWikiLink Dover_Publications.
- Cocountable_topology wikiPageWikiLink Empty_set.
- Cocountable_topology wikiPageWikiLink Hausdorff_space.
- Cocountable_topology wikiPageWikiLink Hyperconnected_space.
- Cocountable_topology wikiPageWikiLink Limit_point_compact.
- Cocountable_topology wikiPageWikiLink Lindelöf_space.
- Cocountable_topology wikiPageWikiLink Locally_connected_space.
- Cocountable_topology wikiPageWikiLink Metacompact_space.
- Cocountable_topology wikiPageWikiLink Open_set.
- Cocountable_topology wikiPageWikiLink Pseudocompact_space.
- Cocountable_topology wikiPageWikiLink Springer-Verlag.
- Cocountable_topology wikiPageWikiLink Springer_Science+Business_Media.
- Cocountable_topology wikiPageWikiLink T1_space.
- Cocountable_topology wikiPageWikiLinkText "Cocountable topology".
- Cocountable_topology wikiPageWikiLinkText "cocountable topology".
- Cocountable_topology wikiPageWikiLinkText "countable complement topology".
- Cocountable_topology hasPhotoCollection Cocountable_topology.
- Cocountable_topology wikiPageUsesTemplate Template:Citation.
- Cocountable_topology subject Category:General_topology.
- Cocountable_topology comment "The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X.Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X. It is also T1, as all singletons are closed.".
- Cocountable_topology label "Cocountable topology".
- Cocountable_topology sameAs Topología_de_los_complementos_numerables.
- Cocountable_topology sameAs Topologie_codénombrable.
- Cocountable_topology sameAs הטופולוגיה_הקו-מנייתית.
- Cocountable_topology sameAs m.02wv7gm.
- Cocountable_topology sameAs Q5139908.
- Cocountable_topology sameAs Q5139908.
- Cocountable_topology wasDerivedFrom Cocountable_topology?oldid=612217974.
- Cocountable_topology isPrimaryTopicOf Cocountable_topology.