Baire Category Theorem

From the Baire Property to Residual Sets

The Baire Category Theorem connects completeness with a powerful principle: in a Baire space, countably many open dense sets have dense intersection. Earlier in this course, we established the theorem and its consequences for complete metric spaces. Here we develop a useful language for applying it. Rather than checking an intersection of open dense sets from scratch each time, we can describe sets as “large” in the category sense and use stable rules for combining them.

Category size is not the same as measure or cardinality. A set can be large in the category sense while having empty interior, and a category argument by itself does not tell us its measure. The definitions below specify exactly what the category terms mean.

The ambient space matters in each definition. For instance, a singleton may be nowhere dense in one space but not in another. The closure and interior used to test whether a set is nowhere dense are always taken in the specified space \(X\).

A Baire space is one in which every countable intersection of open dense sets is dense. The Baire Category Theorem gives this property for every complete metric space. We will use that result as a tool, without repeating its proof.

Residual Sets Contain Dense G-delta Sets

The definition of residual involves a meagre complement, which might not itself be a particularly convenient set to describe. In a Baire space, the Baire Category Theorem gives a more useful characterization: a residual set contains a dense G-delta set. Conversely, containing a dense G-delta set guarantees that a set is residual.

Proof. Suppose first that \(E\) is residual. Then \(X\setminus E\) is meagre, so there are nowhere-dense sets \(A_1,A_2,\ldots\) such that \(X\setminus E\subseteq\bigcup_{n=1}^{\infty}A_n\). For each \(n\), define \(G_n=X\setminus\overline{A_n}\). The set \(G_n\) is open because \(\overline{A_n}\) is closed. It is dense because \(A_n\) is nowhere dense: \(\overline{A_n}\) has empty interior, so its complement meets every nonempty open set.

The Baire property of \(X\) now implies that \(G=\bigcap_{n=1}^{\infty}G_n\) is dense. It is a G-delta set by its definition. If \(x\in G\), then \(x\notin\overline{A_n}\), and in particular \(x\notin A_n\), for every \(n\). Thus \(x\notin X\setminus E\), so \(x\in E\). We have found a dense G-delta set \(G\subseteq E\).

Conversely, suppose \(E\) contains a dense G-delta set \(G=\bigcap_{n=1}^{\infty}U_n\), where every \(U_n\) is open. Since \(G\) is dense and \(G\subseteq U_n\), each \(U_n\) is dense. Therefore \(X\setminus G=\bigcup_{n=1}^{\infty}(X\setminus U_n)\). Each \(X\setminus U_n\) is closed, and its interior is empty because \(U_n\) is dense. Hence each \(X\setminus U_n\) is nowhere dense, so \(X\setminus G\) is meagre. As \(E\) contains \(G\), we have \(X\setminus E\subseteq X\setminus G\), which is meagre. A subset of a meagre set is meagre: if \(M\subseteq\bigcup_n A_n\) with each \(A_n\) nowhere dense, then \(M=\bigcup_n(M\cap A_n)\), and each \(M\cap A_n\) is nowhere dense. It follows that \(X\setminus E\) is meagre, so \(E\) is residual. \(\square\)

The Baire hypothesis is essential for the forward direction: it ensures that the G-delta set constructed from the nowhere-dense pieces is dense. Without that hypothesis, a residual set need not contain a dense G-delta set.

Countable Intersections Preserve Residuality

The dense G-delta characterization gives a convenient way to combine category-large sets. The result below is often the practical form of the Baire Category Theorem used in applications.

Proof. Let \(E_1,E_2,\ldots\) be residual subsets of a Baire space \(X\). By the characterization just proved, for each \(k\) there is a dense G-delta set \(G_k\subseteq E_k\). Write \(G_k=\bigcap_{j=1}^{\infty}U_{k,j}\), where each \(U_{k,j}\) is open and dense. The family of pairs \((k,j)\) is countable, so \(\bigcap_{k=1}^{\infty}G_k=\bigcap_{k=1}^{\infty}\bigcap_{j=1}^{\infty}U_{k,j}\) is a countable intersection of open dense sets. It is dense by the Baire property of \(X\). It is also a G-delta set, and it is contained in \(\bigcap_k E_k\). The residual-set characterization now implies that \(\bigcap_k E_k\) is residual; it is dense because it contains a dense set. \(\square\)

This theorem lets us prove countably many conditions at once. If each condition holds on a residual set, then all the conditions hold simultaneously on a residual set. The conclusion is stronger than saying that each condition holds on a dense set: an intersection of arbitrary dense sets need not be dense, while the Baire property controls intersections of countably many open dense sets.

A Category Argument for Continuous Functions

Category methods are useful not only for describing subsets of familiar spaces but also for proving that a property is typical among functions. A standard setting is \(C([0,1])\), the space of continuous real-valued functions on \([0,1]\), with the supremum metric \(d(f,g)=\|f-g\|_\infty=\sup_{x\in[0,1]}|f(x)-g(x)|\).

This function space is complete. To see why, let \((f_n)\) be Cauchy in the supremum metric. For each \(x\), the real sequence \((f_n(x))\) is Cauchy, since \(|f_n(x)-f_m(x)|\leq\|f_n-f_m\|_\infty\). Define \(f(x)=\lim_n f_n(x)\). The Cauchy property in the supremum metric implies uniform convergence to \(f\): given \(\varepsilon>0\), choose \(N\) such that \(\|f_n-f_m\|_\infty<\varepsilon/2\) for \(n,m\geq N\), and let \(m\to\infty\) to obtain \(|f_n(x)-f(x)|\leq\varepsilon/2<\varepsilon\) for all \(x\) and \(n\geq N\). A uniform limit of continuous functions is continuous, so \(f\in C([0,1])\). Therefore \(C([0,1])\) is complete.

How to Use Category Without Overstating It

A residual set is category-large in a precise sense: its complement is meagre. In a Baire space it also contains a dense G-delta set, and countably many residual properties can be imposed simultaneously. The continuous-function example shows the method: express each condition as an open dense set, check openness and density separately, and then apply the Baire Category Theorem to their intersection.

Keep the scope of the conclusion clear. Density says that every nonempty open set meets a set; it does not say that the set contains an open set. Residuality is defined by a meagre complement; it does not mean that the set is closed or open. Nor does category alone imply a statement about measure. These distinctions prevent a common error: treating “large” as though it had the same meaning in topology, measure theory, and cardinality.