I have the following problem:
Let $J:=(\mathbb{R}/\mathbb{Q})\cap[0,1]$ be the irrational numbers in the intervall $[0,1]$. Show that J can't be written as $J=\bigcup_{n=1}^\infty F_n$ with closed sets $F_n\subset[0,1]$
I guess I can use the "Baire category theorem", which says:
If a complete metric space $X$ is written as an union $X=\bigcup_{n=1}^\infty F_n$ of countable closed sets $F_n$, at least one $F_n$ contains a closed ball.
My problem is now that $J$ isn't complete, because there should be a cauchy-sequence, which converges to $0$, but $0\notin J$. How can I proof my problem without the Baire category theorem?
Another problem is, that the Baire category theorem specifies that the amount of closed sets is countable. Can someone give me a hint how to solve these two problems?
The rest would be simple because $\mathbb{Q}$ is dense in $\mathbb{R}$, so in every ball in a $F_n$ is a rational number and so $J=\bigcup_{n=1}^\infty F_n$ isn't possible. Can someone help me?
Instead of trying to use $J$, use $[0,1]$ as your complete metric space:
Suppose that it is possible to write $J=\bigcup_{n=1}^{\infty}F_n$ as a countable union of closed sets. Each of these sets is nowhere dense because $\mathbb{Q}$ is dense in $\mathbb{R}$. Since $\mathbb{Q}\cap[0,1]$ is countable, we can also write $\mathbb{Q}\cap[0,1]=\bigcup_{n=1}^{\infty}G_n$ as a countable union of closed nowhere dense sets, namely the points of $\mathbb{Q}\cap[0,1]$. Putting these together, it follows that $$ [0,1]=\Big(\bigcup_{n=1}^{\infty}F_n\Big)\cup\Big(\bigcup_{n=1}^{\infty}G_n\Big)$$ is a countable union of closed nowhere dense sets, which contradicts the Baire category theorem.