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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.

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3 Answers 3

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The idea is the same. Instead of choosing $\sqrt{2}$, choose an irrational number in $(0,1)$ and split it to two open intervals.

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Pick $U=A\cap (-\infty,\frac{\sqrt2}{2})$, $V=A\cap (\frac{\sqrt2}{2},\infty)$.

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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.

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