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- Particular_point_topology abstract "In mathematics, the particular point topology (or included point topology) is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionT = {S ⊆ X: p ∈ S or S = ∅}of subsets of X is then the particular point topology on X. There are a variety of cases which are individually named: If X = {0,1} we call X the Sierpiński space. This case is somewhat special and is handled separately. If X is finite (with at least 3 points) we call the topology on X the finite particular point topology. If X is countably infinite we call the topology on X the countable particular point topology. If X is uncountable we call the topology on X the uncountable particular point topology.A generalization of the particular point topology is the closed extension topology. In the case when X \ {p} has the discrete topology, the closed extension topology is the same as the particular point topology.This topology is used to provide interesting examples and counterexamples.".
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- Particular_point_topology wikiPageLength "7870".
- Particular_point_topology wikiPageOutDegree "47".
- Particular_point_topology wikiPageRevisionID "627024896".
- Particular_point_topology wikiPageWikiLink Alexandroff_extension.
- Particular_point_topology wikiPageWikiLink Alexandrov_topology.
- Particular_point_topology wikiPageWikiLink Baire_space.
- Particular_point_topology wikiPageWikiLink Category:Topological_spaces.
- Particular_point_topology wikiPageWikiLink Closed_extension_topology.
- Particular_point_topology wikiPageWikiLink Closure_(topology).
- Particular_point_topology wikiPageWikiLink Compact_space.
- Particular_point_topology wikiPageWikiLink Comparison_of_topologies.
- Particular_point_topology wikiPageWikiLink Connected_space.
- Particular_point_topology wikiPageWikiLink Countable_set.
- Particular_point_topology wikiPageWikiLink Countably_infinite.
- Particular_point_topology wikiPageWikiLink Counterexamples_in_Topology.
- Particular_point_topology wikiPageWikiLink Dense_set.
- Particular_point_topology wikiPageWikiLink Dispersion_point.
- Particular_point_topology wikiPageWikiLink Dover_Publications.
- Particular_point_topology wikiPageWikiLink Excluded_point_topology.
- Particular_point_topology wikiPageWikiLink Extension_topology.
- Particular_point_topology wikiPageWikiLink Finite_set.
- Particular_point_topology wikiPageWikiLink Finite_topological_space.
- Particular_point_topology wikiPageWikiLink First-countable_space.
- Particular_point_topology wikiPageWikiLink Hyperconnected.
- Particular_point_topology wikiPageWikiLink Hyperconnected_space.
- Particular_point_topology wikiPageWikiLink Kolmogorov_space.
- Particular_point_topology wikiPageWikiLink Limit_point.
- Particular_point_topology wikiPageWikiLink Limit_point_compact.
- Particular_point_topology wikiPageWikiLink Lindelöf_space.
- Particular_point_topology wikiPageWikiLink Mathematics.
- Particular_point_topology wikiPageWikiLink Normal_space.
- Particular_point_topology wikiPageWikiLink One-point_compactification.
- Particular_point_topology wikiPageWikiLink Open_set.
- Particular_point_topology wikiPageWikiLink Pseudocompact_space.
- Particular_point_topology wikiPageWikiLink Real_line.
- Particular_point_topology wikiPageWikiLink Regular_space.
- Particular_point_topology wikiPageWikiLink Second-countable_space.
- Particular_point_topology wikiPageWikiLink Separable_space.
- Particular_point_topology wikiPageWikiLink Separated_set.
- Particular_point_topology wikiPageWikiLink Separated_sets.
- Particular_point_topology wikiPageWikiLink Separation_axiom.
- Particular_point_topology wikiPageWikiLink Set_(mathematics).
- Particular_point_topology wikiPageWikiLink Sierpinski_topology.
- Particular_point_topology wikiPageWikiLink Sierpiński_space.
- Particular_point_topology wikiPageWikiLink Springer-Verlag.
- Particular_point_topology wikiPageWikiLink Springer_Science+Business_Media.
- Particular_point_topology wikiPageWikiLink Topological_space.
- Particular_point_topology wikiPageWikiLink Totally_disconnected.
- Particular_point_topology wikiPageWikiLink Totally_disconnected_space.
- Particular_point_topology wikiPageWikiLink Tychonoff_space.
- Particular_point_topology wikiPageWikiLink Ultraconnected.
- Particular_point_topology wikiPageWikiLink Ultraconnected_space.
- Particular_point_topology wikiPageWikiLink Uncountable.
- Particular_point_topology wikiPageWikiLink Uncountable_set.
- Particular_point_topology wikiPageWikiLink Weakly_countably_compact.
- Particular_point_topology wikiPageWikiLink Ω-accumulation_point.
- Particular_point_topology wikiPageWikiLinkText "Countable particular point topology".
- Particular_point_topology wikiPageWikiLinkText "Finite particular point topology".
- Particular_point_topology wikiPageWikiLinkText "Particular point topology".
- Particular_point_topology wikiPageWikiLinkText "Uncountable particular point topology".
- Particular_point_topology wikiPageWikiLinkText "particular point space".
- Particular_point_topology wikiPageWikiLinkText "particular point topology".
- Particular_point_topology hasPhotoCollection Particular_point_topology.
- Particular_point_topology wikiPageUsesTemplate Template:Citation.
- Particular_point_topology subject Category:Topological_spaces.
- Particular_point_topology hypernym Topology.
- Particular_point_topology type Space.
- Particular_point_topology comment "In mathematics, the particular point topology (or included point topology) is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionT = {S ⊆ X: p ∈ S or S = ∅}of subsets of X is then the particular point topology on X. There are a variety of cases which are individually named: If X = {0,1} we call X the Sierpiński space.".
- Particular_point_topology label "Particular point topology".
- Particular_point_topology sameAs m.0dg6ch.
- Particular_point_topology sameAs Точковмісна_топологія.
- Particular_point_topology sameAs Q7140510.
- Particular_point_topology sameAs Q7140510.
- Particular_point_topology wasDerivedFrom Particular_point_topology?oldid=627024896.
- Particular_point_topology isPrimaryTopicOf Particular_point_topology.