Category of a set

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