Corollary to the Baire Category Theorem for Complete Metric Spaces

Recall from The Baire Category Theorem for Complete Metric Spaces that if $X$ is a complete metric space and if $(D_n)_{n=1}^{\infty}$ is a countable collection of open dense sets in $X$ then the intersection

(1)

\begin{align} \quad \bigcap_{n=1}^{\infty} D_n \end{align}

is also a dense set in $X$.

We are about to state an important corollary to the Baire Category theorem for complete metric spaces. We first need the following lemma:

Lemma 1: Let $X$ be a metric space. If $A$ is dense in $X$ then $A^c$ has empty interior.

• Proof: Suppose that $A^c$ does not have empty interior. That is, suppose that:

(2)

\begin{align} \quad \mathrm{int} (A^c) \neq \emptyset \end{align}

• Take any $x \in \mathrm{int} (A^c)$. Since $\mathrm{int} (A^c)$ is an open set there exists a ball centered at $x$ with radius $r > 0$ such that $B(x, r) \subseteq \mathrm{int} (A^c)$. But $B(x, r) \subseteq A^c$. But then $B(x, r) \cap A \subseteq A^c \cap A = \emptyset$. This contradicts $A$ being dense in $X$. So the assumption that $\mathrm{int} (A^c) \neq \emptyset$ was false. Hence:

(3)

\begin{align} \quad \mathrm{int} (A^c) = \emptyset \end{align}

Corollary 2: Let $X$ be a complete metric space. If $(F_n)_{n=1}^{\infty}$ is any countable collection of closed and nowhere dense sets in $X$ then $\displaystyle{\bigcup_{n=1}^{\infty} F_n}$ has empty interior.

In the proof below we use the following result: A set is nowhere dense if and only if the complement of the closure of that set is dense. So a closed set $C$ is nowhere dense if and only if $C^c$ is dense.

• Proof: Let $(F_n)_{n=1}^{\infty}$ be a countable collection of closed and nowhere dense sets in $X$. Then $(F_n^c)_{n=1}^{\infty}$ is a countable collection of open sets that are dense in $X$. By the Baire Category theorem we have that:

(4)

\begin{align} \quad \bigcap_{n=1}^{\infty} F_n^c \end{align}

• is also a dense set in $X$. By De Morgan's Laws:

(5)

\begin{align} \quad \left ( \bigcup_{n=1}^{\infty} F_n \right )^c \end{align}

• is a dense set in $X$. But this implies that the complement:

(6)

\begin{align} \quad \left ( \left ( \bigcup_{n=1}^{\infty} F_n \right )^c \right )^c = \bigcup_{n=1}^{\infty} F_n \end{align}

• has empty interior by Lemma 1. $\blacksquare$