Category of a set

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 $X$ Otherwise $E$ is said to be of the second category (such sets are also called meager). 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, (cf. [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 ([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)