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:X→Y with pointwise limit f:X→Y. 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.
Read more about this topic: Baire Space
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)