Baire spacetopology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are "enough" points for certain limit processes.
Definition: A Baire space is a topological space X that satisfies one (and therefore all) of the following equivalent conditions:
A common proof technique in analysis is the following: one first shows that the given space X is Baire (typically using general theorems mentioned below), and then one applies condition 3 in order to show that certain interior points must exist.
Examples of Baire spaces:
- All complete metric spaces are Baire spaces (this is the Baire category theorem).
- Every space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space (this includes the irrational numbers with their standard topology, as well as the Cantor set).
- Every locally compact Hausdorff space is a Baire space (this includes all manifolds).
Two closely related definitions often appear, especially in older literature:
Definition: A subset of a topological space X is meagre in X (or of first category in X) if it is a union of countably many nowhere dense subsets of X. A subset of X which is not meagre is called of second category in X.
(Note that this notion of "category" has nothing to do with category theory.)
In this language, a topological space X is a Baire space if and only if every non-empty open set is of second category in X. In particular, 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 [0,1].
B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space: the product of countably many copies of the discrete space N. This is a Baire space in the above topological sense. As a topological space, B is homeomorphic to the set Ir of irrational numbers carrying their standard topology inherited from the reals. The homeomorphism between B and Ir can be constructed using continued fractions. The uniform structures of B and Ir are different however: B is complete and Ir is not.
Baire space should be contrasted with Cantor space, the set of infinite sequences of binary digits.