Baire category turns topology into a largeness principle: in complete or locally compact settings, countable intersections of dense open sets remain dense.
Nowhere dense and meagre sets
Category distinguishes topologically small sets from large residual sets.
Definition. A subset \(A\subseteq X\) is nowhere dense if \((\overline A)^\circ=\varnothing\). It is meagre, or of first category, if it is a countable union of nowhere dense sets. A residual set is the complement of a meagre set.
Definition. A Baire space is a space in which every countable intersection of dense open sets is dense.
Proposition. A space is Baire if and only if no nonempty open set is meagre in itself.
Proof. If \(\bigcap_nU_n\) is not dense for dense open \(U_n\), then some nonempty open \(W\) is contained in \(\bigcup_n(X\setminus U_n)\), a countable union of nowhere dense closed sets relative to \(W\). Conversely, if a nonempty open \(W\) is meagre, write \(W=\bigcup_nA_n\) with \(A_n\) nowhere dense in \(X\) after taking relative closures; then the dense open sets \(X\setminus\overline{A_n}\) have intersection missing \(W\).
Complete metric theorem
Completeness prevents dense open requirements from collapsing under countable intersection.
Theorem. Every complete metric space is a Baire space.
Proof. Let \(U_n\) be dense open sets and let \(O\) be a nonempty open set. Choose a closed ball \(\overline B_1\subseteq O\cap U_1\) of radius at most \(1\). Inductively choose closed balls \(\overline B_{n+1}\subseteq B_n\cap U_{n+1}\) with radius at most \(2^{-n}\). The centers form a Cauchy sequence, hence converge to some \(x\). Nested closed balls force \(x\in\overline B_n\subseteq U_n\) for every \(n\), and \(x\in O\). Thus \(O\) meets \(\bigcap_nU_n\).
Locally compact Hausdorff theorem
Local compactness is the nonmetric route to Baire category.
Theorem. Every locally compact Hausdorff space is a Baire space.
Proof. Given dense open \(U_n\) and nonempty open \(O\), choose a nonempty open \(V_1\) with compact closure inside \(O\cap U_1\). Inductively choose nonempty open \(V_{n+1}\) with compact closure inside \(V_n\cap U_{n+1}\). The compact closed sets \(\overline{V_n}\) have the finite intersection property, so their intersection is nonempty. Any point in the intersection lies in \(O\cap\bigcap_nU_n\).
Consequences in analysis
Baire arguments often prove existence by showing the bad set is meagre.
Corollary. A complete metric space cannot be written as a countable union of closed sets with empty interior.
Proof. Closed sets with empty interior are nowhere dense. A complete metric space is Baire, so the whole space, if nonempty, is not meagre in itself.
Example. The uniform boundedness principle and open mapping theorem in functional analysis are Baire category theorems disguised as linear analysis.
Remark. Topological largeness by category differs from measure-theoretic largeness: a set can be dense \(G_\delta\) and have measure zero.