Baire category theorem

The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense).

The theorem was proved by French mathematician René-Louis Baire in his 1899 doctoral thesis.

Statement

A Baire space is a topological space with the property that for each countable collection of open dense sets (Un)∞
n=1, their intersection ∩n ∈ ℕ Un is dense.

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.

Relation to the axiom of choice

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.[2]

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.[3] This restricted form applies in particular to the real line, the Baire space ωω, the Cantor space 2ω, and a separable Hilbert space such as L2(ℝn).

Uses

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:

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.

Proof

The following is a standard proof that a complete pseudometric space is a Baire space.

Let Un be a countable collection of open dense subsets. We want to show that the intersection ∩ Un 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 W in X has a point x in common with all of the Un. Since U1 is dense, W intersects U1; thus, there is a point x1 and 0 < r1 < 1 such that:

B(x1, r1) ⊆ W ∩ U1

where B(x, r) and B(x, r) denote an open and closed ball, respectively, centered at x with radius r. Since each Un is dense, we can continue recursively to find a pair of sequences xn and 0 < rn < 1/n such that:

B(xn, rn) ⊆ B(xn-1, rn-1) ∩ Un.

(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 xn.) Since xn ∈ B(xm, rn) when n > m, we have that xn is Cauchy, and hence xn converges to some limit x by completeness. For any n, by closedness, x ∈ B(xn, rn).

Therefore, x ∈ W and x ∈ Un for all n.

There is an alternative proof by M. Baker for the proof of the theorem using Choquet's game.[4]

gollark: Maybe I need an optimized Rust program which does this.
gollark: `base64 </dev/urandom | rg cats` is able to do it quite fast.
gollark: `/dev/urandom` exists, you can look at it.
gollark: Besides, they can't, it's coming on stdin from `cat`.
gollark: I don't believe so.

See also

Notes

    Citations
    1. Narici & Beckenstein 2011, pp. 371–423.
    2. Blair 1977.
    3. Levy 2002, p. 212.
    4. Baker 2014.

    Works cited
    • Baire, R. (1899). "Sur les fonctions de variables réelles". Ann. di Mat. 3: 1–123.
    • Baker, Matt (July 7, 2014). "Real Numbers and Infinite Games, Part II: The Choquet game and the Baire Category Theorem". Matt Baker's Math Blog.
    • 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) [First published 1979]. Basic Set Theory (Reprinted ed.). Dover. ISBN 0-486-42079-5.
    • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-158488866-6. OCLC 144216834.
    • 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).

    Further reading
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