Baire Space - Examples

Examples

  • The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself. The rational numbers are of first category and the irrational numbers are of second category in R.
  • The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval with the usual topology.
  • Here is an example of a set of second category in R with Lebesgue measure 0.
where is a sequence that enumerates the rational numbers.
  • Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.

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