In an arbitrary topological space, the class of closed sets with emptyinterior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets in, smooth curves in the plane, and proper affine subspaces in a Euclidean space. If a topological space is a Baire space then it is "large", meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space is not a countable union of its affine planes.
Definition
The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. First, we give the usual modern definition, and then we give a historical definition that is closer to the definition originally given by Baire.
Definitions
In his original definition, Baire defined a notion of category as follows. Note that a closed subset is nowhere dense if and only if its interior is empty.
Baire space definition
Sufficient conditions
Baire category theorem
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an important tool in topology and functional analysis.
A topological vector space is nonmeagre if and only if it is a Baire space, which happens if and only if every closed absorbing subset has non-empty interior.
Examples
The space 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.
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.
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.
Non-example
One of the first non-examples comes from the induced topology of the rationals inside of the real line with the standard euclidean topology. Given an indexing of the rationals by the natural numbers, so a bijection, and let where, which is an open, dense subset in. Then, because the intersection of every open set in is empty, the space cannot be a Baire space.
Properties
Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of 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.
A closed subset of a Baire space is not necessarily Baire.
The product of two Baire spaces is not necessarily Baire. However, there exist sufficient conditions that will guarantee that a product of arbitrarily many Baire spaces is again Baire.
