(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

  1. In R, any singleton (i.e., a set consisting of a single point) is nowhere dense. So any countable set is of first category.
  2. In particular, the rationals Q are a countable set.
  3. 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.
  4. 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.
  5. 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!)

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