Show that $[0,1] \cap \mathbb{Q}$ is neither connected nor closed

Answer

(1) Not connected

Note that $[0,1] \cap \mathbb{Q} = [0,\sqrt{2}/2) \cup (\sqrt{2}/2,1]$ and $[0,\sqrt{2}/2),(\sqrt{2}/2,1]$ are separated.

(2) Not compact

Consider the open cover $\{(-1,\sqrt{2}/2-1/n) \cup (\sqrt{2}/2+1/n,2)\}_{n=2}^\infty$. For any finite subcover $\{(-1,\sqrt{2}/2-1/n_k) \cup (\sqrt{2}/2+1/n_k,2)\}_{k=1}^N$, there exists $q \in N_{1/n_\ast}(\sqrt{2}/2) \cap \mathbb{Q}$ such that $q \notin \bigcup_{k=1}^N(-1,\sqrt{2}/2-1/n_k) \cup (\sqrt{2}/2+1/n_k,2)$ where $n_\ast>\max(n_1,\cdots,n_N)$.

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