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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](http://math.stackexchange.com/questions/26311/).2011-12-06

2 Answers 2

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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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Sure. Part a) is not a "problem" but a famous theorem, the Baire Category Theorem. Note that the linked to wikipedia article gives a proof.

(By giving this answer, I am acting on my opinion that it is not reasonable to expect a student to come up with a proof of this on her own, or at least that a reasonable response to being asked to do so is to look up the proof.)

Hint for part b): take complements.

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    I’m torn: as a teacher I tend to agree, especially nowadays with so much readily available and easily found. But two of the best and most enjoyable courses that I ever took taught using the Moore method. I was introduced to topology as a freshman by John Greever, using his book before it was published, and Jim Cannon used it even more strictly when he taught the graduate general topology course at Madison in ’69-’70.2011-12-06