Two proofs with possibly Baire category theorem about completness.

Question

I'm working with completness right now and I've come across two interesting problems. In my opinion they are worth a little bit attention .

a) Let $K$ be closed subset with empty interior on euclidean line. Prove that $\exists \ {t \in \mathbb{R}}$ : set $ \{ t + x : x \in K \}$ is contained in the set of irrationals.

b) Let $\{F_1, F_2,...\}$ be closed set with empty interior on euclidean line $\mathbb{R}$, and $A \subset \mathbb{R}$ be countable set. Prove that $\exists \ {t \in \mathbb{R}}$ : set $ \bigcup_i F_i$ is disjont with $ \{ t + a : a \in A \}$.

Well, I really think that Baire category theorem is my starting point. I will try to follow that path and show you some results later on. In meanwhile I would really appreciate if you posted your observations, so if I were mistaken I would notice that as soon as possible.

Answer 1

You’re right that the Baire category theorem is involved. Here’s a hint for (a). Suppose that there is no such $t\in\Bbb R$. Then for each $t\in\Bbb R$ there is a $q_t\in\Bbb Q\cap(t+K)$. In other words, there is an $x_t\in K$ such that $t+x_t=q_t$, and therefore $t=q_t-x_t\in q_t-K$. Use this to show that $\Bbb R$ is the union of countably many closed, nowhere dense sets and so get a contradiction.

Added: I’ve added a big hint for (b) in the spoiler-protected block below. It really is just a slight slight elaboration of the ideas of (a), with almost nothing new.

Let $F=\bigcup_{n\in\Bbb Z^+}F_n$. Suppose that $F\cap(t+A)\ne\varnothing$ for each $t\in\Bbb R$. Then for each $t\in\Bbb R$ there are an $n(t)\in\Bbb Z^+$ and an $a_t\in A$ such that $t+a_t\in F_{n(t)}$, i.e., $t\in F_{n(t)}-a_t$. Now consider $\{F_n-a:n\in\Bbb Z^+\text{ and }a\in A\}$.