A countable union of nowhere dense sets is a set of first category. Not as imaginatively named as the second category!
These 2 categories are most commonly used in complete metric spaces -- they are part of the formulation of the Baire category theorem.
- In R, any singleton (i.e., a set consisting of a single point) is nowhere dense. So any countable set is of first category.
- In particular, the rationals Q are a countable set.
- The Cantor ternary set is nowhere dense, so it is of first category. Note that this is an example of an uncountable set of first category.
- In the same way, it turns out that for any digit sequence d1d2...dk the set of all numbers in [0,1) whose decimal expansion does not contain d1d2...dk is nowhere dense.
- Since there are only countably many finite digit sequences d1d2...dk, the set of all numbers in [0,1) whose decimal expansions do not contain all finite digit sequences is of first category. (However it is dense!)