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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 $R$.

M.Krov
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2 Answers2

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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.

Leitingok
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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.

Pete L. Clark
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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 06 '11 at 04:33
  • Well, perhaps my parenthetical comment was worded too strongly: of course I agree that it depends on what has already been done. But in general I am not a Moore-ist: I feel like the task of the instructor and/or the text is to prove the theorems presented (or, failing that, to indicate where these proofs can be found). Assigning standard theorems as exercises seems somewhat lazy to me: surely one can come up with other exercises for the students to try to solve, such that failure to solve them correctly does not leave a gaping hole in their education. – Pete L. Clark Dec 06 '11 at 11:50
  • 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. – Brian M. Scott Dec 06 '11 at 16:36