Cocountability: Difference between revisions
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{{Short description| |
{{Short description|Having all but countably many elements}} |
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| ⚫ | In [[mathematics]], a '''cocountable''' [[subset]] of a set ''X'' is a subset ''Y'' whose [[complement (set theory)|complement]] in ''X'' is a [[countable set]]. In other words, ''Y'' contains all but countably many elements of ''X''. Since the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says ''Y'' is [[cofinite]].{{r|halgiv}} |
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{{unreferenced|date=May 2021}} |
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| ⚫ | In [[mathematics]], a '''cocountable''' [[subset]] of a set ''X'' is a subset ''Y'' whose [[complement (set theory)|complement]] in ''X'' is a [[countable set]]. In other words, ''Y'' contains all but countably many elements of ''X''. Since the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says ''Y'' is [[cofinite]]. |
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== σ-algebras == |
== σ-algebras == |
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The set of all subsets of ''X'' that are either countable or cocountable forms a [[sigma-algebra|σ-algebra]], i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the '''countable-cocountable algebra''' on ''X''. It is the smallest σ-algebra containing every [[singleton set]]. |
The set of all subsets of ''X'' that are either countable or cocountable forms a [[sigma-algebra|σ-algebra]], i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the '''countable-cocountable algebra''' on ''X''. It is the smallest σ-algebra containing every [[singleton set]].{{r|sigma}} |
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== Topology == |
== Topology == |
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The [[cocountable topology]] (also called the "countable complement topology") on any set ''X'' consists of the [[empty set]] and all cocountable subsets of ''X''. |
The [[cocountable topology]] (also called the "countable complement topology") on any set ''X'' consists of the [[empty set]] and all cocountable subsets of ''X''.{{r|james}} |
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==References== |
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{{reflist|refs= |
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<ref name=halgiv>{{citation |
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| last1 = Halmos | first1 = Paul |
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| last2 = Givant | first2 = Steven |
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| contribution = Chapter 5: Fields of sets |
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| doi = 10.1007/978-0-387-68436-9_5 |
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| isbn = 9780387684369 |
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| location = New York |
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| pages = 24–30 |
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| publisher = Springer |
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| series = Undergraduate Texts in Mathematics |
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| title = Introduction to Boolean Algebras |
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| year = 2009}}</ref> |
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<ref name=james>{{citation |
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| last = James | first = Ioan Mackenzie |
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| doi = 10.1007/978-1-4471-3994-2 |
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| isbn = 9781447139942 |
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| journal = Springer Undergraduate Mathematics Series |
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| location = London |
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| page = 33 |
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| publisher = Springer |
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| title = Topologies and Uniformities |
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| year = 1999}}</ref> |
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<ref name=sigma>{{harvtxt|Halmos|Givant|2009}}, "Chapter 29: Boolean σ-algebras", pp. 268–281, {{doi|10.1007/978-0-387-68436-9_29}}</ref> |
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}} |
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[[Category:Basic concepts in infinite set theory]] |
[[Category:Basic concepts in infinite set theory]] |
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Latest revision as of 01:06, 8 April 2024
In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. Since the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says Y is cofinite.[1]
σ-algebras
[edit]The set of all subsets of X that are either countable or cocountable forms a σ-algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the countable-cocountable algebra on X. It is the smallest σ-algebra containing every singleton set.[2]
Topology
[edit]The cocountable topology (also called the "countable complement topology") on any set X consists of the empty set and all cocountable subsets of X.[3]
References
[edit]- ^ Halmos, Paul; Givant, Steven (2009), "Chapter 5: Fields of sets", Introduction to Boolean Algebras, Undergraduate Texts in Mathematics, New York: Springer, pp. 24–30, doi:10.1007/978-0-387-68436-9_5, ISBN 9780387684369
- ^ Halmos & Givant (2009), "Chapter 29: Boolean σ-algebras", pp. 268–281, doi:10.1007/978-0-387-68436-9_29
- ^ James, Ioan Mackenzie (1999), "Topologies and Uniformities", Springer Undergraduate Mathematics Series, London: Springer: 33, doi:10.1007/978-1-4471-3994-2, ISBN 9781447139942
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