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In the mathematical fields of [[general topology]] and [[descriptive set theory]], a '''meagre set''' (also called a '''meager set''' or a '''set of first category''') is a set that, considered as a [[subset]] of a (usually larger) [[topological space]], is in a precise sense small or [[negligible set|negligible]]. The meagre subsets of a fixed space form a [[sigma-ideal]] of subsets; that is, any subset of a meagre set is meagre, and the [[union (mathematics)|union]] of [[countable set|countably]] many meagre sets is meagre.
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General topologists use the term [[Baire space]] to refer to a broad class of topological spaces on which the notion of meagre set is not trivial (in particular, the entire space is not meagre). [[Descriptive set theory|Descriptive set theorists]] mostly study meagre sets as subsets of the [[real number]]s, or more generally any [[Polish space]], and reserve the term [[Baire space (set theory)|Baire space]] for one particular Polish space.
 
The [[complement (set theory)|complement]] of a meagre set is a '''comeagre set''' or '''residual set'''.
 
== Definition ==
Given a topological space ''X'', a subset ''A'' of ''X'' is meagre if it can be expressed as the union of countably many ''[[nowhere dense]]'' subsets of ''X''.  [[Duality (mathematics)|Dually]], a comeagre set is one whose [[complement (set theory)|complement]] is meagre, or equivalently, the [[intersection (set theory)|intersection]] of countably many sets with dense interiors.
 
A subset ''B'' of ''X'' is '''nowhere dense''' if there is no [[neighborhood (mathematics)|neighbourhood]] on which ''B'' is [[dense set|dense]]: for any nonempty open set ''U'' in ''X'', there is a nonempty open set ''V'' contained in ''U'' such that ''V'' and ''B'' are [[disjoint sets|disjoint]].
 
The complement of a nowhere dense set is a dense set.  More precisely, the complement of a nowhere dense set is a set with ''dense [[interior (topology)|interior]]''.  Not every dense set has a nowhere dense complement.  The complement of a dense set can have nowhere dense, and dense regions.
 
=== Relation to Borel hierarchy ===
Just as a nowhere dense subset need not be closed, but is always contained in a closed nowhere dense subset (viz, its closure), a meagre set need not be an [[Fσ set|F<sub>σ</sub> set]] (countable union of closed sets), but is always contained in an F<sub>σ</sub> set made from nowhere dense sets (by taking the closure of each set).
 
Dually, just as the complement of a nowhere dense set need not be open, but has a dense [[interior (topology)|interior]] (contains a dense open set), a comeagre set need not be a [[Gδ set|G<sub>δ</sub> set]] (countable intersection of [[open set|open]] sets), but contains a dense G<sub>δ</sub> set formed from dense open sets.
 
==Terminology==
A meagre set is also called a  ''set of first category'';  a nonmeagre set (that is, a set that is not meagre) is also called a ''set of second category''. Second category does ''not'' mean comeagre – a set may be neither meagre nor comeagre (in this case it will be of second category).
 
==Properties==
* Any subset of a meagre set is meagre; any superset of a comeagre set is comeagre.
* The union of countable many meagre sets is also meagre; the intersection of countably many comeagre sets is comeagre.
::This follows from the fact that a countable union of countable sets is countable.
 
==Banach–Mazur game==
Meagre sets have a useful alternative characterization in terms of the [[Banach–Mazur game]]. If <math>Y</math> is a topological space, <math>W</math> is a family of subsets of <math>Y</math> which have nonempty interior such that every nonempty open set has a subset in <math>W</math>, and <math>X</math> is any subset of <math>Y</math>, then there is a Banach-Mazur game corresponding to <math>X, Y, W</math>. In the Banach-Mazur game, two players, <math>P_1</math> and <math>P_2</math>, alternate choosing successively smaller (in terms of the subset relation) elements of <math>W</math> to produce a descending sequence <math>W_1 \supset W_2 \supset W_3 \supset \dotsb</math>. If the intersection of this sequence contains a point in <math>X</math>, <math>P_1</math> wins; otherwise, <math>P_2</math> wins. If <math>W</math> is any family of sets meeting the above criteria, then <math>P_2</math> has a [[winning strategy]] if and only if <math>X</math> is meagre.
 
==Examples==
=== Subsets of the reals ===
* The [[rational numbers]] are meagre as a subset of the reals and as a space – that is, they do not form a [[Baire space]].
* The [[Cantor set]] is meagre as a subset of the reals, but ''not'' as a space, since it is a complete metric space and is thus a [[Baire space]], by the [[Baire category theorem]].
 
=== Function spaces ===
* The set of functions which have a derivative at some point is a meagre set in the space of all [[continuous function]]s.<ref>{{cite journal|author=Banach, S.|title=Über die Baire'sche Kategorie gewisser Funktionenmengen|journal=Studia. Math.|issue=3|year=1931|pages=174–179}}</ref>
 
== Notes ==
<div class="references-small" style="-moz-column-count:2; column-count:2;">
<references />
</div>
 
== See also ==
* [[Baire category theorem]]
* [[Generic property]], for analogs to residual
* [[Negligible set]], for analogs to meagre
 
== External links ==
 
* [http://mathoverflow.net/questions/43478/is-there-a-measure-zero-set-which-isnt-meagre Is there a measure zero set which isn’t meagre?]
 
[[Category:General topology]]
[[Category:Descriptive set theory]]

Latest revision as of 19:14, 24 November 2014

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