Corollary to the Baire Category Theorem for Complete Metric Spaces

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. $\blacksquare$
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