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 | + | 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 $X$ (such sets are also called meager). Otherwise $E$ is said to be of the second category. |
| − | + | This notation 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). 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. {{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 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 (Lebesgue) measure, while there are (Lebesgue) null sets which are residual ({{Cite|vR}}, Th. 5.5). | |
| − | (cf. {{Cite|Ro}}) | ||
====References==== | ====References==== | ||
Revision as of 16:01, 8 August 2012
2020 Mathematics Subject Classification: Primary: 54A05 [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 $X$ (such sets are also called meager). Otherwise $E$ is said to be of the second category. This notation 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). 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 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 (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) |
| [Ox] | J.C. Oxtoby, "Measure and category" , Springer (1971) |
| [Ro] | H.L. Royden, "Real analysis", Macmillan (1968) |
| [vR] | A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press (1982) |
Category of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Category_of_a_set&oldid=27307