In mathematics, a topological space is said to be a Baire space, if for any given countable collection of closed sets with empty interior in , their union also has empty interior in . Equivalently, a locally convex space which is not meagre in itself is called a Baire space. According to Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of a Baire space. Bourbaki coined the term "Baire space".
In an arbitrary topological space, the class of closed sets with empty interior 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.
The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. A topological space is called a Baire space if it satisfies any of the following equivalent conditions:
- Every intersection of countably many dense open sets in is dense in ;
- Every union of countably many closed subsets of with empty interior has empty interior;
- Every non-empty open subset of is a nonmeager subset of ;
- Every comeagre subset of is dense in ;
- Whenever the union of countably many closed subsets of has an interior point, then at least one of the closed subsets must have an interior point;
- Every point in has a neighborhood that is a Baire space (according to any defining condition other than this one).
- So is a Baire space if and only if it is "locally a Baire space."
Baire category theoremEdit
- (BCT1) Every complete pseudometric space is a Baire space. More generally, every topological space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, every completely metrizable space is a Baire space.
- (BCT2) Every locally compact Hausdorff space (or more generally every locally compact sober space) is a Baire space.
BCT1 shows that each of the following is a Baire space:
- The space of real numbers
- The space of irrational numbers, which is homeomorphic to the Baire space of set theory
- Every compact Hausdorff space is a Baire space.
- In particular, the Cantor set is a Baire space.
- Indeed, every Polish space.
Other sufficient conditionsEdit
- 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 .
- Another large class of Baire spaces are algebraic varieties with the Zariski topology. For example, the space of complex numbers whose open sets are complements of the vanishing sets of polynomials is an algebraic variety with the Zariski topology. Usually this is denoted .
- 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 with Lebesgue measure :
- Note that the space of rational numbers with the usual topology inherited from the real numbers is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.
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.
- 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.
- Given a family of continuous functions = with pointwise limit If is a Baire space then the points where is not continuous is a meagre set in and the set of points where is continuous is dense in A special case of this is the uniform boundedness principle.
- 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.
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