Baire category theorem

A Baire space is a topological space with the property that for each countable collection of open dense sets ${\displaystyle \{U_{n}\}_{n=1}^{\infty }}$, their intersection ${\displaystyle \textstyle \bigcap _{n=1}^{\infty }U_{n}}$ is dense.

• (BCT1) Every complete metric space is a Baire space. Thus every completely metrizable topological 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.
• (BCT2) Every locally compact Hausdorff space is a Baire space. The proof is similar to the preceding statement; the finite intersection property takes the role played by completeness.

Neither of these statements directly implies the other, since there are complete metric spaces that are not locally compact (the irrational numbers with the metric defined below; also, any Banach space of infinite dimension), and there are locally compact Hausdorff spaces that are not metrizable (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in functional analysis; the uncountable Fort space). See Steen and Seebach in the references below.

• (BCT3) A non-empty complete metric space, or any of its subsets with nonempty interior, is not the countable union of nowhere-dense sets.

This formulation is equivalent to BCT1 and is sometimes more useful in applications. Also: if a non-empty complete metric space is the countable union of closed sets, then one of these closed sets has non-empty interior.

The proof of BCT1 for arbitrary complete metric spaces requires some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to the axiom of dependent choice, a weak form of the axiom of choice.[1]

A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be separable, is provable in ZF with no additional choice principles.[2] This restricted form applies in particular to the real line, the Baire space ω^ω, the Cantor space 2^ω, and a separable Hilbert space such as L²(Rⁿ).

BCT1 is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.

BCT1 also shows that every complete metric space with no isolated points is uncountable. (If X is a countable complete metric space with no isolated points, then each singleton {x} in X is nowhere dense, and so X is of first category in itself.) In particular, this proves that the set of all real numbers is uncountable.

BCT1 shows that each of the following is a Baire space:

• The space ${\displaystyle \mathbb {R} }$ of real numbers
• The irrational numbers, with the metric defined by ${\displaystyle d(x,y)={\tfrac {1}{n+1}}}$, where ${\displaystyle n}$ is the first index for which the continued fraction expansions of ${\displaystyle x}$ and ${\displaystyle y}$ differ (this is a complete metric space)
• The Cantor set

By BCT2, every finite-dimensional Hausdorff manifold is a Baire space, since it is locally compact and Hausdorff. This is so even for non-paracompact (hence nonmetrizable) manifolds such as the long line.

BCT is used to prove Hartogs's theorem, a fundamental result in the theory of several complex variables.

The following is a standard proof that a complete pseudometric space ${\displaystyle \scriptstyle X}$ is a Baire space.

Let ${\displaystyle U_{n}}$ be a countable collection of open dense subsets. We want to show that the intersection ${\displaystyle \bigcap U_{n}}$ is dense. A subset is dense if and only if every nonempty open subset intersects it. Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set ${\displaystyle W}$ in ${\displaystyle X}$ has a point ${\displaystyle x}$ in common with all of the ${\displaystyle U_{n}}$. Since ${\displaystyle U_{1}}$ is dense, ${\displaystyle W}$ intersects ${\displaystyle U_{1}}$; thus, there is a point ${\displaystyle x_{1}}$ and ${\displaystyle 0\;<\;r_{1}\;<\;1}$ such that:

${\displaystyle {\overline {B}}(x_{1},r_{1})\subseteq W\cap U_{1}}$

where ${\displaystyle B(x,r)}$ and ${\displaystyle {\overline {B}}(x,r)}$ denote an open and closed ball, respectively, centered at ${\displaystyle x}$ with radius ${\displaystyle r}$. Since each ${\displaystyle U_{n}}$ is dense, we can continue recursively to find a pair of sequences ${\displaystyle x_{n}}$ and ${\displaystyle 0\;<\;r_{n}\;<\;{\frac {1}{n}}}$ such that:

${\displaystyle {\overline {B}}(x_{n},r_{n})\subset B(x_{n-1},r_{n-1})\cap U_{n}}$

(This step relies on the axiom of choice and the fact that a finite intersection of open sets is open and hence an open ball can be found inside it centered at ${\displaystyle x_{n}}$) Since ${\displaystyle x_{n}\;\in \;B(x_{m},r_{m})}$ when ${\displaystyle n\;>\;m}$, we have that ${\displaystyle x_{n}}$ is Cauchy, and hence ${\displaystyle x_{n}}$ converges to some limit ${\displaystyle x}$ by completeness. For any ${\displaystyle n}$, by closedness,

${\displaystyle x\in {\overline {B}}(x_{n},r_{n}).}$

Therefore, ${\displaystyle x\;\in \;W}$ and ${\displaystyle x\;\in \;U_{n}}$ for all ${\displaystyle n}$.

There is an alternative proof by M. Baker for the proof of the theorem using Choquet's game.[3]

• Property of Baire

• Baire, R. (1899). "Sur les fonctions de variables réelles". Ann. di Mat. 3: 1–123.
• Blair, Charles E. (1977). "The Baire category theorem implies the principle of dependent choices". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (10): 933–934.
• Gamelin, Theodore W.; Greene, Robert Everist. Introduction to Topology (2nd ed.). Dover.
• Levy, Azriel (2002) [1979]. Basic Set Theory (Reprinted ed.). Dover. ISBN 0-486-42079-5.
• Schechter, Eric. Handbook of Analysis and its Foundations. Academic Press. ISBN 0-12-622760-8.
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology. New York: Springer-Verlag. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).

• Tao, T. "245B, Notes 9: The Baire category theorem and its Banach space consequences".
• Encyclopaedia of Mathematics article on Baire theorem