# dense subsets in metric spaces

I am stuck on this problem. Any help?

Problem

a)- Let $X$ be a complete metric space and let $V_{n}$, $n=1,2,3,...$ be open and dense sets. Prove that $\bigcap_{n=1}^{\infty }V_{n}$ is dense in $X$ .

b)- Use part a) to prove that the set of irrational numbers cannot be written as a union of countably many closed subsets of $R$.

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For part b) see also here. –  t.b. Dec 6 '11 at 4:26

Part a) is a famous theorem, the Baire Category Theorem.

b) Suppose $\mathbb R -\mathbb Q =\bigcup_{n\in \mathbb N} F_n$,where $F_n$ is closed for every $n$.so we can write $\mathbb Q$ as $\bigcap_{n\in \mathbb N}(\mathbb R -F_n)$ . Note that every $\mathbb R -F_n$ is open and contains $\mathbb Q$, therefore dense.

Let $\mathbb Q = \{q_1,q_2...\}=\bigcup_{n\in \mathbb N} \{q_n\}$

then $\emptyset=\mathbb Q- \mathbb Q = \bigcap_{n\in \mathbb N}(\mathbb R -F_n-\{q_n\} )$. Note that the set on the right side of the equality is intersection of countable many open dense sets. By a) it must be dense. Contradiction.

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Depends on what’s already been done. If they’ve seen the nested balls theorem and the proof for the compact $T_2$ case, it’s trivial; if they’ve just seen the former, it’s still pretty easy; if they’ve just seen the latter, it’s at the hard end of reasonable. Greever’s modified Moore method text leaves it to the student, but he’s previously given a proof that co-countable subsets of $\mathbb{R}$ are second countable and suggests using it as a model. –  Brian M. Scott Dec 6 '11 at 4:33