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Difference between revisions of "Category of a set"

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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
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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 set|nowhere dense sets]] in $X$, otherwise $E$ is said to be of the second category (cp. with Chapter 9 of {{Cite|Ox}}).  
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(such sets are also called meager).  
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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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Sometimes the complement in $X$ of a set of the first category is also called a set of the second category. However, in the case of a [[Baire space]] a more appropriate name for such sets is residual (or comeager). 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 and 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).
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(cf. {{Cite|Ro}}). 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, in $\mathbb R$ a set of the first category can be a set of full measure, while there are null sets which are residual ({{Cite|vR}}, Th. 5.5).
 
  
 
====References====
 
====References====
 
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{|
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|valign="top"|{{Ref|Ba}}|| R. Baire,  "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars  (1905)
 
 
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer (1971)
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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|Ro}}|| H.L. Royden, "Real analysis", Macmillan (1968)
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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|vR}}|| A.C.M. van Rooy,  W.H. Schikhof,  "A second course on real functions" , Cambridge Univ. Press  (1982)
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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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Revision as of 08:24, 18 August 2013

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 0217.09201 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=27307
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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