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A topological characterization of the  "massiveness"  of a set. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020760/c0207601.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020760/c0207602.png" /> is said to be of the first category in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020760/c0207603.png" /> if it can be expressed as a finite or countable union of nowhere-dense sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020760/c0207604.png" />. Otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020760/c0207605.png" /> is said to be of the second category. Sometimes the complement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020760/c0207606.png" /> of a set of the first category is also called a set of the second category. In modern literature (see [[#References|[2]]]) such sets (in the case of a [[Baire space|Baire space]]) are called residual or comeager. A non-empty closed set of real numbers, in particular an interval, is not of the first category in itself [[#References|[1]]]. This result generalizes to any complete metric space. This generalization has wide application in analysis. The role of a set of the first category in analysis is analogous to that of a null set in measure theory. However, a set of the first category can be a set of full measure, while there are null sets of the second category.
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[[Category:Topology]]
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====References====
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A topological characterization of the  "massivenessof a set. A subset $E$ of a topological space $X$ is said to be of the first category in $X$ if it can be expressed as a finite or countable union of [[nowhere-dense set|nowhere dense sets]] in $X$, otherwise $E$ is said to be of the second category (cp. with Chapter 9 of {{Cite|Ox}}).
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Baire,  "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.C. Oxtoby,   "Measure and category" , Springer  (1971)</TD></TR></table>
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This terminology is, however, not universal: some authors use the name second category for complements in $X$ of sets of the first category. In the case of a [[Baire space]] a more appropriate name for such sets is residual (or comeager), cp. again with Chapter 9 of {{Cite|Ox}}. A non-empty closed set of real numbers, in particular an interval, is not of the first category in itself. This result generalizes to any complete metric space, it is called [[Baire category theorem]]
 
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(cf. {{Cite|Ro}}) and has wide application in analysis (a primary example is the [[Banach-Steinhaus theorem]]). The role of a set of the first category in topology is analogous to that of a null set in measure theory. However, in $\mathbb R$ a set of the first category can be a set of full (Lebesgue) measure, while there are (Lebesgue) null sets which are residual ({{Cite|vR}}, Th. 5.5).
 
 
 
 
====Comments====
 
A set of the first category is also called a meager set. The statement that a complete metric space is not of the first category in itself is known as Baire's category theorem (cf. [[#References|[a1]]]).
 
 
 
Examples of sets that are meager and not null, as well as of those that are null and not meager, can be found in [[#References|[a2]]], Th. 5.5.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.L. Royden,   "Real analysis" , Macmillan  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"A.C.M. van Rooy,   W.H. Schikhof,   "A second course on real functions" , Cambridge Univ. Press  (1982)</TD></TR></table>
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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}}
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby, "Measure and category" , Springer (1971) {{MR|0393403}} {{ZBL|0217.09201}}
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|valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis", Macmillan  (1968) {{MR|0151555}} {{ZBL|0197.03501}}
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|valign="top"|{{Ref|vR}}|| A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press  (1982) {{MR|655599}}
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Latest revision as of 19:06, 7 December 2023

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

A topological characterization of the "massiveness" of a set. A subset $E$ of a topological space $X$ is said to be of the first category in $X$ if it can be expressed as a finite or countable union of nowhere dense sets in $X$, otherwise $E$ is said to be of the second category (cp. with Chapter 9 of [Ox]). This terminology is, however, not universal: some authors use the name second category for complements in $X$ of sets of the first category. In the case of a Baire space a more appropriate name for such sets is residual (or comeager), cp. again with Chapter 9 of [Ox]. A non-empty closed set of real numbers, in particular an interval, is not of the first category in itself. This result generalizes to any complete metric space, it is called Baire category theorem (cf. [Ro]) and has wide application in analysis (a primary example is the Banach-Steinhaus theorem). The role of a set of the first category in topology is analogous to that of a null set in measure theory. However, in $\mathbb R$ a set of the first category can be a set of full (Lebesgue) measure, while there are (Lebesgue) null sets which are residual ([vR], Th. 5.5).

References

[Ba] R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) Zbl 36.0438.01
[Ox] J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 Zbl 0217.09201
[Ro] H.L. Royden, "Real analysis", Macmillan (1968) MR0151555 Zbl 0197.03501
[vR] A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press (1982) MR655599
How to Cite This Entry:
Category of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Category_of_a_set&oldid=15117
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article