Baire space

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

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 $ℝ$ , 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. First, we give some preliminary definitions that Baire used in his original work, then we'll introduce some modern equivalent definitions.

Definitions

In his original definition, Baire defined a notion of category (unrelated to category theory) as follows.

Definition: A subset of a topological space is called nowhere dense or rare if its closure has empty interior.

Note that a closed subset is nowhere dense if and only if its interior is empty.

Definition: A subset of a topological space

X

is said to be meagre in

X

, a meagre subset of

X

, or of the first category in

X

if it is a countable union of nowhere dense subsets of

X

. A subset is of the second category or nonmeagre in

X

if it is not of first category in

X

.

Definition: A topological space is called meagre (resp. nonmeagre) if it is a meagre (resp. nonmeagre) subset of itself.

Warning: Note that if

S

is a subset of

X

then when we say that

S

is a meagre subspace of

X

then we mean that when

S

is endowed with the subspace topology (induced by

X

) then

S

is a meagre topological space (i.e.

S

is a meagre subset of

S

). In contrast, if we say that

S

is a meagre subset of

X

then we mean that it is equal to a countable union of nowhere dense subsets of

X

. The same applies to nonmeager subsets and subspaces.

Definition: A subset

A

of

X

is comeagre in

X

if its complement

X ∖ A

is meagre in

X

.

Baire space definition

Definition — A topological space $X$ is called a Baire space if it satisfies any of the following equivalent conditions:

Every non-empty open subset of $X$ is a nonmeager subset of $X$ ;[1]
Every comeagre subset of $X$ is dense in $X$ ;
The union of any countable collection of closed nowhere dense subsets (i.e. each closed subset has empty interior) has empty interior;[1]
Every intersection of countably many dense open sets in X is dense in X;[1]
- In comparison, in every topological space, the intersection of any finite collection of dense open subsets is again a dense open subset.
The interior (taken in $X$ ) of every union of countably many closed nowhere dense sets is empty;
Whenever the union of countably many closed subsets of $X$ has an interior point, then at least one of the closed subsets must have an interior point;
The complement in $X$ of every meagre subset of $X$ is dense in $X$ ;[1]
Every point in X has a neighborhood that is a Baire space (according to any defining condition other than this one).[1]
- So $X$ 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) Every complete pseudometric space is a Baire space.[1] 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 $ℝ$ of real numbers
The space of irrational numbers, which is homeomorphic to the Baire space $ω ω$ 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.[1]
A topological vector space is nonmeagre if and only if it is a Baire space,[1] which happens if and only if every closed absorbing subset has non-empty interior.[2]

Examples

The space $ℝ$ 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 $ℝ$ .
The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval $[0, 1]$ with the usual topology.
Here is an example of a set of second category in $ℝ$ with Lebesgue measure 0:
$\bigcap _{m=1}^{\infty }\bigcup _{n=1}^{\infty }\left(r_{n}-({\tfrac {1}{2}})^{n+m},r_{n}+({\tfrac {1}{2}})^{n+m}\right)$
where $(r n) \infty n =1$ is a sequence that enumerates the rational numbers.
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 $ℚ$ inside of the real line $ℝ$ with the standard euclidean topology. Given an indexing of the rationals by the natural numbers $ℕ$ , so a bijection $f : ℕ \to ℚ$ , and let $𝒜 = (A n) \infty n =1$ where $A n = ℚ ∖ {f (n)$ }, which is an open, dense subset in $ℚ$ . Then, because the intersection of every open set in $𝒜$ is empty, the space $ℚ$ 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 $X$ 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 $[0, 1]$ .
Every open subspace of a Baire space is a Baire space.
Given a family of continuous functions $f n : X \to Y$ with pointwise limit $f : X \to Y$ . If $X$ is a Baire space then the points where $f$ is not continuous is a meagre set in $X$ and the set of points where $f$ is continuous is dense in $X$ . 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.

References

Narici & Beckenstein 2011, pp. 371-423.
Wilansky 2013, p. 60.

Bibliography

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, Topology, 2nd edition, Prentice Hall, 2000.
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 (1969). 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.
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.

External links

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[FOOTNOTENariciBeckenstein2011371-423-1] Narici & Beckenstein 2011, pp. 371-423.

[FOOTNOTEWilansky201360-2] Wilansky 2013, p. 60.