# Proving that the set of rational numbers between 0 and 1 is disconnected.

Show that the set of rational numbers between 0 and 1 $$(A = \mathbb{Q} \cap [0,1])$$ is disconnected. Note that $$A \subseteq \mathbb{R}$$ is a subspace topology.

Definition of disconnectedness:

Suppose that $$X$$ has a topology $$T$$. We say that $$X$$ is disconnected if there are two non-empty open sets $$U, V \in T$$ such that $$U \cap V = \emptyset$$ and $$U \cup V = X$$.

If the question did not specify a specific interval, I would have picked $$U = (-\infty,\sqrt{2})$$ and $$V = (\sqrt{2}, \infty)$$, and proceeded to show that all $$\mathbb{Q}$$ is disconnected. I do not have an idea to prove it between 0 and 1 using only the definition.

The idea is the same. Instead of choosing $$\sqrt{2}$$, choose an irrational number in $$(0,1)$$ and split it to two open intervals.
Pick $$U=A\cap (-\infty,\frac{\sqrt2}{2})$$, $$V=A\cap (\frac{\sqrt2}{2},\infty)$$.
Take $$P=[0,1/\pi)\cap\mathbb Q$$ and $$Q=(1/\pi,1]\cap\mathbb Q$$ such that $$P\cup Q=A$$. Now $$P\cap\overline Q=\phi$$ and $$\overline P\cap Q=\phi$$ suffices to prove the result.