# Baire space

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In mathematics, a topological space ${\displaystyle X}$ is said to be a Baire space, if for any given countable collection ${\displaystyle \{A_{n}\}}$ of closed sets with empty interior in ${\displaystyle X}$, their union ${\displaystyle \cup A_{n}}$ also has empty interior in ${\displaystyle X}$. [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 ${\displaystyle \mathbb {R} ,}$ 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 ${\displaystyle X}$ is called a Baire space if it satisfies any of the following equivalent conditions:

1. Every non-empty open subset of ${\displaystyle X}$ is a nonmeager subset of ${\displaystyle X}$; [5]
2. Every comeagre subset of ${\displaystyle X}$ is dense in ${\displaystyle X}$;
3. The union of any countable collection of closed nowhere dense subsets (i.e. each closed subset has empty interior) has empty interior; [5]
4. Every intersection of countably many dense open sets in ${\displaystyle X}$ is dense in ${\displaystyle X}$; [5]
5. The interior (taken in ${\displaystyle X}$) of every union of countably many closed nowhere dense sets is empty;
6. Whenever the union of countably many closed subsets of ${\displaystyle X}$ has an interior point, then at least one of the closed subsets must have an interior point;
7. The complement in ${\displaystyle X}$ of every meagre subset of ${\displaystyle X}$ is dense in ${\displaystyle X}$; [5]
8. Every point in ${\displaystyle X}$ has a neighborhood that is a Baire space (according to any defining condition other than this one). [5]
• So ${\displaystyle X}$ 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 shows that each of the following is a Baire 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 ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle \mathbb {R} }$.
• The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval ${\displaystyle [0,1]}$ with the usual topology.
• Here is an example of a set of second category in ${\displaystyle \mathbb {R} }$ with Lebesgue measure ${\displaystyle 0}$:
${\displaystyle \bigcap _{m=1}^{\infty }\bigcup _{n=1}^{\infty }\left(r_{n}-({\tfrac {1}{2}})^{n+m},r_{n}+({\tfrac {1}{2}})^{n+m}\right)}$
where ${\displaystyle \left(r_{n}\right)_{n=1}^{\infty }}$ is a sequence that enumerates the rational numbers.
• 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 ${\displaystyle \mathbb {Q} }$ inside of the real line ${\displaystyle \mathbb {R} }$ with the standard euclidean topology. Given an indexing of the rationals by the natural numbers ${\displaystyle \mathbb {N} }$ so a bijection ${\displaystyle f:\mathbb {N} \to \mathbb {Q} ,}$ and let ${\displaystyle {\mathcal {A}}=\left(A_{n}\right)_{n=1}^{\infty }}$ where ${\displaystyle A_{n}:=\mathbb {Q} \setminus \{f(n)\},}$ which is an open, dense subset in ${\displaystyle \mathbb {Q} .}$ Then, because the intersection of every open set in ${\displaystyle {\mathcal {A}}}$ is empty, the space ${\displaystyle \mathbb {Q} }$ 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 ${\displaystyle X}$ 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 ${\displaystyle [0,1].}$
• Every open subspace of a Baire space is a Baire space.
• Given a family of continuous functions ${\displaystyle f_{n}:X\to Y}$= with pointwise limit ${\displaystyle f:X\to Y.}$ If ${\displaystyle X}$ is a Baire space then the points where ${\displaystyle f}$ is not continuous is a meagre set in ${\displaystyle X}$ and the set of points where ${\displaystyle f}$ is continuous is dense in ${\displaystyle X.}$ 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.

## Citations

1. Munkres 2000, p. 295.
2. Köthe 1979, p. 25.
3. Munkres 2000, p. 296.
4. Haworth & McCoy 1977, p. 5.
5. Narici & Beckenstein 2011, pp. 371-423.
6. Wilansky 2013, p. 60.

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## References

• Baire, René-Louis (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 33, 1–123.
• Grothendieck, Alexander (1973). . 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