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The Baire Category Theorem

In this section we introduce the Baire category theorem. Ultimately the Baire category theorem is used to answer the question when is a subset E⊂XE \subset X in a topological space small, which turns out to be a fundamental tools in the study of linear bounded operators. This section and in particular the qualiatative and quantitative discussion is inspired by T. Tao Tao (2009).

Small sets in a Measure Space

In a measure theory setting i.e. X=(X,F,μ)X = (X, \mathcal{F}, \mu) the “small” sets are the nullsets i.e. E⊂XE \subset X such that μ(E)=0\mu(E) = 0 or subsets of nullsets. Countable additivity implies that countable unions of nullsets are of measure zero. This gives us results like this:

Lemma 1

Let E1,E2,…E_1, E_2, \dots be an at most countable sequence of measurable sets of a measure XX. If μ(⋃nEn)>0\mu\left( \bigcup_n E_n \right) > 0 then at least one of the EnE_n has positive measure.

The next proposition bridges between measure theory and topology. To motivate it, recall that the Lebesgue differentiation theorem tells us that for almost every point x∈Ex \in E, the density of EE at xx,

lim⁡r→0μ(E∩B(x,r))μ(B(x,r))=1,\lim_{r \to 0} \frac{\mu(E \cap B(x,r))}{\mu(B(x,r))} = 1,
(1)

equals one. Such points are called density points of EE — points where EE fills up nearly all of every sufficiently small ball. One can think of density points as measure-theoretic interior points: an interior point of EE is one where EE contains an entire ball, while a density point is one where EE nearly fills every small ball, even if it doesn’t contain one outright. The next proposition globalizes this local statement: if EE has positive measure, then EE must be dense in some ball in the measure-theoretic sense that it occupies an arbitrarily large fraction of that ball’s volume.

Proposition 1

Let EE be a measurable subset of Rd\mathbb{R}^d. Then the following are equivalent

  1. μ(E)>0\mu(E) > 0

  2. ∀ε>0\forall \varepsilon > 0 there exists a ball BB such that μ(E∩B)≥(1−ε)μ(B)\mu(E \cap B) \geq (1 - \varepsilon)\mu(B)

Proof:

We prove Proposition 1 in two steps.

  • (1)  ⟹  (2)(1) \implies (2). We proceed by contradiction. Suppose that μ(E)>0\mu(E) > 0 but there exists ε0>0\varepsilon_0 > 0 such that every ball satisfies μ(E∩B)<(1−ε0)μ(B)\mu(E \cap B) < (1 - \varepsilon_0) \mu(B)

    The Lebesgue differentiation theorem implies that for almost every x∈Ex\in E we have that lim⁡r→0μ(E∩B(x,r))μ(B(x,r))=1\lim_{r \to 0} \frac{\mu(E \cap B(x, r))}{\mu(B(x, r))} = 1

    Since EE has positive measure there must be at least one point where the Lebesgue differentiation theorem applies. For this we can find a sufficiently small rr so that μ(E∩B(x,r))μ(B(x,r))>1−ε0.\frac{\mu(E \cap B(x, r))}{\mu(B(x, r))} > 1 - \varepsilon_0.

    A contradiction.

  • (2)  ⟹  (1)(2) \implies (1). Let ε=1/2\varepsilon = 1/2, then there exists a ball BB such that μ(E∩B)≥12μ(B)\mu(E \cap B) \geq \frac{1}{2}\mu(B)

    Since we have μ(B)>0\mu(B) > 0 we have μ(E∩B)>0\mu(E \cap B) > 0, and we have that μ(E)≥μ(E∩B)>0.\mu(E) \geq \mu(E \cap B) > 0.

Reading the equivalence from both sides:

A set of measure zero is therefore nowhere dense in the measure-theoretic sense: it is never concentrated in any ball. This suggests that to formalize “smallness” in a purely topological setting, we should look for the topological analogue of this property.

Small Sets in Topological Spaces

To make the analogy with the measure-theoretic setting precise, we first recall what dense means in a topological space.

Definition 1 (Dense in an open set)

A set EE in a topological space is dense in an open set UU if every nonempty open subset of UU intersects EE, or equivalently, if the closure of EE contains UU.

Remark 1

Compare this with the measure-theoretic notion from the previous section: there, EE is “dense in BB” if μ(E∩B)≈μ(B)\mu(E \cap B) \approx \mu(B), i.e. EE fills almost all the volume of BB. Here, EE is dense in BB if EE meets every open subset of BB, i.e. EE is topologically everywhere in BB. Volume has been replaced by topology, but the intuition is the same: EE is thoroughly present throughout BB, with no region where EE is absent.

Definition 2 (Nowhere dense)

A set EE is nowhere dense if it is not dense in any ball, or equivalently, if its closure has empty interior.

This is the topological counterpart of a measure-zero set: just as a null set is never measure-theoretically dense in any ball (it never fills a large fraction of any ball), a nowhere dense set is never topologically dense in any ball (its closure contains no open set).

Theorem 1 (Baire Category Theorem)

Let E1,E2,…E_1, E_2, \dots be an at most countable sequence of subsets of a complete metric space XX. If ⋃nEn\bigcup_n E_n contains a ball BB, then at least one of the EnE_n is dense in a sub-ball B′⊂BB' \subset B (and in particular not nowhere dense). To put it in the contrapositive: the countable union of nowhere dense sets cannot contain a ball.

Note that completeness is essential: on an incomplete metric space, the Baire property can fail, and the space itself may be a countable union of nowhere dense sets. This is why completeness is such an important hypothesis throughout the theory of bounded operators. For a proof, see Hillen (2023).

Definition 3 (Meagre (First Category))

A set EE in a topological space is meagre (or of the first category) if it can be written as a countable union of nowhere dense sets. A set that is not meagre is called non-meagre (or of the second category).

Remark 2

A meagre set need not be nowhere dense itself. For example, Q⊂R\mathbb{Q} \subset \mathbb{R} is meagre — it is the countable union ⋃q∈Q{q}\bigcup_{q \in \mathbb{Q}} \{q\} of nowhere dense singletons — but Q\mathbb{Q} is not nowhere dense, since Q‾=R\overline{\mathbb{Q}} = \mathbb{R} has nonempty interior. Thus “meagre” is a strictly weaker notion than “nowhere dense”: every nowhere dense set is meagre (a union of one nowhere dense set), but not conversely.

Corollary 1 (A complete metric space is not meagre)

Let XX be a complete metric space and {Fj}\{F_j\} a countable collection of nowhere dense sets. Then ⋃jFj≠X\bigcup_{j} F_j \neq X. Equivalently, a complete metric space is not meagre (i.e. not a countable union of nowhere dense sets).

Example 1 (The rationals are meagre but not nowhere dense)

On X=RX = \mathbb{R} (which is complete), the rationals Q=⋃q∈Q{q}\mathbb{Q} = \bigcup_{q \in \mathbb{Q}} \{q\} are a countable union of nowhere dense sets, hence meagre. But Q\mathbb{Q} is not nowhere dense: Q‾=R\overline{\mathbb{Q}} = \mathbb{R}, which has nonempty interior. Corollary 1 tells us Q≠R\mathbb{Q} \neq \mathbb{R}, which we know. More importantly: the irrationals R∖Q\mathbb{R} \setminus \mathbb{Q} are a dense GδG_\delta set (a countable intersection of open dense sets), and Q\mathbb{Q} cannot be GδG_\delta.

Example 2 (The Cantor set is meagre and nowhere dense)

The middle-thirds Cantor set C⊂[0,1]C \subset [0,1] is closed and has empty interior (it contains no interval), so it is nowhere dense. Since every nowhere dense set is trivially meagre (a “union” of one nowhere dense set), CC is also meagre. Yet CC is uncountable — so a set can be topologically small (nowhere dense, meagre) while being large in cardinality. Contrast this with Q\mathbb{Q}, which is meagre but not nowhere dense.

Example 3 (Why do we care?)

The consequence of this theorem is that Rd\mathbb{R}^d or Cd\mathbb{C}^d cannot be covered by a countable union of proper subspaces.

Definition 4 (Proper subspaces)

W⊂VW \subset V is a proper subspace of VV such that

  1. A subset VV i.e. W⊂VW \subset V.

  2. Strictly smaller than VV.

  3. Still a vector space.

In R3\mathbb{R}^3 examples of proper subspaces are:

  1. The xyxy-plane.

  2. A line through the origin.

Baire says that we can’t write R3\mathbb{R}^3 as a countable union of these nowhere dense proper subspaces.

\mathbb{R}^2 cannot be covered by a countable union of proper subspaces (lines through the origin).

Figure 1:R2\mathbb{R}^2 cannot be covered by a countable union of proper subspaces (lines through the origin).

Remark 3

Note that of course we could also prove these results using measure theory in finite dimensional spaces by using countable additivity of countable unions. The advantage of Baire’s theorem is that it easily extends to infinite dimensional vectorspaces.

Remark 4

The Baire Category Theorem holds in two important classes of topological spaces:

  1. Complete metric spaces (and hence Banach spaces, Hilbert spaces, etc.).

  2. Locally compact Hausdorff spaces (and hence Rd\mathbb{R}^d, Cd\mathbb{C}^d, and all finite-dimensional normed spaces).

In finite dimensions these overlap: Rd\mathbb{R}^d is both complete and locally compact. In infinite dimensions, however, a Banach space is never locally compact (the closed unit ball is not compact), so completeness is the only route to the Baire property.

Consequences of the Baire Category Theorem

The Baire Category Theorem leads to three fundamental equivalences between the qualitative theory of continuous linear operators on Banach spaces e.g. finiteness, surjectivity to the quantitative theory e.g. estimates Tao (2009). We have already seen the first of these relationships. Recall that for a linear operator A:X↦YA : X \mapsto Y we have that A is continuous  ⟺   A is bounded.A\ \mathrm{is\ continuous} \iff\ A\ \mathrm{is\ bounded}.

As we saw with the examples this allows us to prove the continuity of an operator AA by simply establishing that it is bounded (which usually is easier).

  1. Uniform boundedness principle that equates the qualitative operators with their quantitative boundedness.

  2. Open mapping theorem equates qualitative solvability of a linear problem Lu=fLu = f with quantitative solvability.

  3. Closed graph theorem equates the qualitative regularity of a (weakly continuous) operator TT with the quantitative regularity of the operator.

Remark 5

Note that these results are not used much in practice, because one usually works in the reverse direction i.e. first prove bounds, and then derive the operator’s qualitative properties. But crucially these results provide the motivation or justification of why we approach qualitative problems in functional analysis via their quantitative counter parts.

Next we will explore the uniform boundedness principle, and the open mapping theorem while we leave the closed graph theorem for later.

References
  1. Tao, T. (2009, February 1). 245B, Notes 9: The Baire category theorem and its Banach space consequences [Blog post]. https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/
  2. Hillen, T. (2023). Elements of Applied Functional Analysis. https://www.math.ualberta.ca/~thillen/FA-book-June2023.pdf
Applied Functional Analysis
Bounded Operators
Applied Functional Analysis
The uniform boundedness principle (Banach-Steinhaus)