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Difference between revisions of "Baire property"

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{{MSC|54E52}}
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[[Category:Topology]]
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''of a subset of a topological space''
  
''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150401.png" /> in a topological space''
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A property analogous to the measurability of a set. A subset $A$ of a topological space $X$ is said to have the Baire property if there is an open set $U$ such that the symmetric difference $(U\setminus A)\cup (A\setminus U)$ is a [[Category of a set|set of first category]], i.e. it is a countable union of [[Nowhere dense set|nowhere dense sets]]. Cp. with Chapter 4 of {{Cite|Ox}}. Some authors use the terminology ''almost open''.
  
A property analogous to the measurability of a set. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150402.png" /> has the Baire property if there exists an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150403.png" /> such that the differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150405.png" /> are sets of the first category according to Baire (cf. [[Category of a set|Category of a set]]); the term  "open"  may be replaced by  "closed" . There are other equivalent definitions; thus, for instance, a set has the Baire property if it is the union of a set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150406.png" /> and a set of the first category. The class of sets with the Baire property is closed under the operations of complementing, taking countable unions and taking countable intersections. For an example of a set which does not have the Baire property, see [[#References|[1]]].
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The following are useful characterizations.
  
====Comments====
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'''Theorem 1'''
A set with the Baire property is often called a Baire set or an almost-open set. A set of the first category is often called meager.
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A set $A$ has the Baire property if and only if there is a closed set $C$ such that $(C\setminus A)\cup (A\setminus C)$ is of first category, cp. with Theorem 4.1 of {{Cite|Ox}}.
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'''Theorem 2'''
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A set $A$ has the Baire property if and only if it is the union of a [[G-delta|$G_\delta$]] and a set of first category (resp. if and only if it there is a set of first category $B$ such that $A\cup B$ is an [[F-sigma|$F_{\sigma}$]]). Cp. with Theorem 4.4 of {{Cite|Ox}}.
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It follows easily from the definition that any countable union of sets with the Baire property has the Baire property. Moreover, it follows from Theorem 1 that the complement of a set with the Baire property has the Baire property. Therefore, the family of subsets of a topological space $X$ with the Baire property is a [[Algebra of sets|$\sigma$-algebra]].
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Sets with the Baire property can also be represented as differences of ''regular open sets'' and sets of first category (an open set is called ''regular'' if it coincides with the interior of its closure).
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Moreover, on subsets of $\mathbb R$ with the Baire property the [[Banach-Mazur game]] is determined (cp. with Theorem 6.3 of {{Cite|Ox}}).
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The axiom of choice implies the existence of subsets of the real numbers without the Baire property: the [[Non-measurable set|Vitali set]] is a classical example.
  
 
====References====
 
====References====
 
{|
 
{|
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|valign="top"|{{Ref|Ba}}||  R. Baire,  "Leçons sur les fonctions discontinues, professées au  collège de France" , Gauthier-Villars  (1905) {{ZBL|36.0438.01}}
 
|valign="top"|{{Ref|Ba}}||  R. Baire,  "Leçons sur les fonctions discontinues, professées au  collège de France" , Gauthier-Villars  (1905) {{ZBL|36.0438.01}}
 
|-
 
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|valign="top"|{{Ref|Ku}}|| K. Kuratowski,  "Topology" , '''1–2''' , Acad. Press (1966–1968) {{MR|0259835}} {{MR|0217751}} {{ZBL|0158.40802}} {{ZBL|0148.42801}} {{ZBL|0104.27402}} {{ZBL|0098.24110}} {{ZBL|0137.15604}} {{ZBL|0041.09604}} {{ZBL|0041.09603}} {{ZBL|0008.13202}}  {{ZBL|56.0517.04}}
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|valign="top"|{{Ref|Ba}}|| E. Čech,  "Topological spaces" , Wiley (1966)  {{MR|0211373}} {{ZBL|0141.39401}}
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|-
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|valign="top"|{{Ref|Ku}}|| K. Kuratowski,  "Topology" , '''1–2''' , Acad. Press (1966–1968) {{MR|0259835}} {{MR|0217751}} {{ZBL|0158.40802}}
 
|-
 
|-
 
|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer  (1971) {{MR|0393403}} {{ZBL| 0217.09201}}
 
|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer  (1971) {{MR|0393403}} {{ZBL| 0217.09201}}
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,  "Topological spaces" , Wiley  (1966)  {{MR|0211373}} {{ZBL|0141.39401}} </TD></TR></table>
 

Latest revision as of 13:42, 7 October 2012

2020 Mathematics Subject Classification: Primary: 54E52 [MSN][ZBL]

of a subset of a topological space

A property analogous to the measurability of a set. A subset $A$ of a topological space $X$ is said to have the Baire property if there is an open set $U$ such that the symmetric difference $(U\setminus A)\cup (A\setminus U)$ is a set of first category, i.e. it is a countable union of nowhere dense sets. Cp. with Chapter 4 of [Ox]. Some authors use the terminology almost open.

The following are useful characterizations.

Theorem 1 A set $A$ has the Baire property if and only if there is a closed set $C$ such that $(C\setminus A)\cup (A\setminus C)$ is of first category, cp. with Theorem 4.1 of [Ox].

Theorem 2 A set $A$ has the Baire property if and only if it is the union of a $G_\delta$ and a set of first category (resp. if and only if it there is a set of first category $B$ such that $A\cup B$ is an $F_{\sigma}$). Cp. with Theorem 4.4 of [Ox].

It follows easily from the definition that any countable union of sets with the Baire property has the Baire property. Moreover, it follows from Theorem 1 that the complement of a set with the Baire property has the Baire property. Therefore, the family of subsets of a topological space $X$ with the Baire property is a $\sigma$-algebra.

Sets with the Baire property can also be represented as differences of regular open sets and sets of first category (an open set is called regular if it coincides with the interior of its closure).

Moreover, on subsets of $\mathbb R$ with the Baire property the Banach-Mazur game is determined (cp. with Theorem 6.3 of [Ox]).

The axiom of choice implies the existence of subsets of the real numbers without the Baire property: the Vitali set is a classical example.

References

[Ba] R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) Zbl 36.0438.01
[Ba] E. Čech, "Topological spaces" , Wiley (1966) MR0211373 Zbl 0141.39401
[Ku] K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) MR0259835 MR0217751 Zbl 0158.40802
[Ox] J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 0217.09201 Zbl 0217.09201
[Ro] H.L. Royden, "Real analysis", Macmillan (1968) MR0151555 Zbl 0197.03501
How to Cite This Entry:
Baire property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_property&oldid=28325
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article