#### (In a metric space:)

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.

### Examples

- 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 d
_{1}d_{2}...d_{k}the set of all numbers in [0,1) whose decimal expansion does not contain d_{1}d_{2}...d_{k}is nowhere dense. - Since there are only
*countably many*finite digit sequences d_{1}d_{2}...d_{k}, 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!)