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 .^{[1]} Equivalently, a locally convex space which is not meagre in itself is called a Baire space.^{[2]} According to Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of a Baire space.^{[3]} Bourbaki coined the term "Baire space".^{[4]}

## Motivation

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.

## Definition

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 non-empty open subset of is a nonmeager subset of ;
^{[5]} - Every comeagre subset of is dense in ;
- The union of any countable collection of closed nowhere dense subsets (i.e. each closed subset has empty interior) has empty interior;
^{[5]} - Every intersection of countably many dense open sets in is dense in ;
^{[5]} - The interior (taken in ) of every union of countably many closed nowhere dense sets is empty;
- 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;
- The complement in of every meagre subset of is dense in ;
^{[5]} - Every point in has a neighborhood that is a Baire space (according to any defining condition other than this one).
^{[5]}- So is a Baire space if and only if it is "locally a Baire space."

## 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.

- (
**BCT1**) Every complete pseudometric space is a Baire space.^{[5]}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.

**BCT2** shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable.
For example, the long line is of second category.

### Other sufficient conditions

- A product of complete metric spaces is a Baire space.
^{[5]} - A topological vector space is nonmeagre if and only if it is a Baire space,
^{[5]}which happens if and only if every closed absorbing subset has non-empty interior.^{[6]}

## 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.
- 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.

### 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.
- 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.

## See also

- Baire space (set theory)
- Banach–Mazur game
- Barrelled space
- Descriptive set theory
- Meagre set
- Nowhere dense set
- Property of Baire

## Citations

**^**Munkres 2000, p. 295.**^**Köthe 1979, p. 25.**^**Munkres 2000, p. 296.**^**Haworth & McCoy 1977, p. 5.- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}Narici & Beckenstein 2011, pp. 371-423. **^**Wilansky 2013, p. 60.

## References

- Baire, René-Louis (1899), Sur les fonctions de variables réelles,
*Annali di Mat. Ser. 3***3**, 1–123. - Grothendieck, Alexander (1973).
*Topological Vector Spaces*. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. - Munkres, James R. (2000).
*Topology*. Prentice-Hall. ISBN 0-13-181629-2. - Khaleelulla, S. M. (1982).
*Counterexamples in Topological Vector Spaces*. Lecture Notes in Mathematics.**936**. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. - Köthe, Gottfried (1983).
*Topological Vector Spaces I*. Grundlehren der mathematischen Wissenschaften.**159**. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. - Köthe, Gottfried (1979).
*Topological Vector Spaces II*. Grundlehren der mathematischen Wissenschaften.**237**. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972. - Rudin, Walter (1991).
*Functional Analysis*. International Series in Pure and Applied Mathematics.**8**(Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. - Wilansky, Albert (2013).
*Modern Methods in Topological Vector Spaces*. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. - Haworth, R. C.; McCoy, R. A. (1977),
*Baire Spaces*, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk

## External links

- Encyclopaedia of Mathematics article on Baire space
- Encyclopaedia of Mathematics article on Baire theorem