About: Particular point topology

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In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is the particular point topology on X. There are a variety of cases that are individually named: 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.

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rdfs:label	Particular point topology (en) Точковмісна топологія (uk)
rdfs:comment	Нехай X — непорожня множина і , де — фіксована точка. Тоді τ є топологією на X, яка називається точковмісною. (uk) In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is the particular point topology on X. There are a variety of cases that are individually named: 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. (en)
dcterms:subject	Topological spaces
Wikipage page ID	5336200 (xsd:integer)
Wikipage revision ID	1103767612 (xsd:integer)
Link from a Wikipage to another Wikipage	Homeomorphic Ω-accumulation point Lindelöf space Compact space Constant function Countably infinite Mathematics Normal space Sierpiński space Separable space Pseudocompact space Tychonoff space Second-countable space Closure (topology) Alexandrov topology Function (mathematics) Connected space Totally disconnected Closed set Comparison of topologies Dense-in-itself Dense set Empty set Topological spaces Weakly countably compact Disjoint sets Dispersion point Finite set Isolated point Kolmogorov space Regular space Baire space Counterexamples in Topology Cover (topology) Accumulation point Discrete topology Dover Publications Continuous function (topology) Hyperconnected Interior (topology) Neighborhood (mathematics) Open set Set (mathematics) Separation axiom Union (set theory) List of topologies Limit point Subspace topology Excluded point topology Finite topological space First-countable space Real line Injective Topological space Overlapping interval topology Subset Ultraconnected Uncountable Uncountable set Preimage Springer-Verlag One-point compactification Sierpinski topology Closed extension topology Separated set Locally relatively compact
sameAs	Particular point topology Particular point topology Particular point topology Particular point topology Particular point topology
dbp:wikiPageUsesTemplate	dbt:Citation
has abstract	In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is the particular point topology on X. There are a variety of cases that are individually named: * If X has two points, the particular point topology on X is the Sierpiński space. * If X is finite (with at least 3 points), the topology on X is called the finite particular point topology. * If X is countably infinite, the topology on X is called the countable particular point topology. * If X is uncountable, the topology on X is called 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. (en) Нехай X — непорожня множина і , де — фіксована точка. Тоді τ є топологією на X, яка називається точковмісною. (uk)
gold:hypernym	Topology
prov:wasDerivedFrom	wikipedia-en:Particular_point_topology?oldid=1103767612&ns=0
page length (characters) of wiki page	8782 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Particular_point_topology
is Link from a Wikipage to another Wikipage of	Compact space Sierpiński space Pseudocompact space Relatively compact subspace Dispersion point Locally Hausdorff space Locally compact space Paracompact space Kolmogorov space

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