Baire Space - Properties

Properties

  • Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval .
  • Every open subspace of a Baire space is a Baire space.
  • Given a family of continuous functions fn:XY with pointwise limit f:XY. If X is a Baire space then the points where f is not continuous is a meagre set in X and the set of points where f is continuous is dense in X. A special case of this is the uniform boundedness principle.

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