Is $[0,1] ∩ Q$ open or closed in real numbers?

Question

According to https://en.wikipedia.org/wiki/Closed_set, " $(0,1) ∩ Q^c $ is not closed in the real numbers". Why is it so?

I think that its complement is $(-∞,0)∪ (1,∞) ∪ Q^c$ is open and all points of $(0,1) ∩ Q^c $ are within or on the boundary, so I think $(0,1) ∩ Q^c $ is closed.

Is $[0,1] ∩ Q$ also open?

Answer 1

It is neither open nor closed:

• $0\in[0,1]\cap\mathbb Q$, but $[0,1]\cap\mathbb Q$ contains no interval $(-\varepsilon,\varepsilon)$ and therefoere $[0,1]\cap\mathbb Q$ is not open;
• you can take a sequence $(x_n)_{n\in\mathbb N}$ of elements of $[0,1]\cap\mathbb Q$ which converges to, say $\sqrt{\frac12}$, but $\sqrt{\frac12}\notin[0,1]\cap\mathbb Q$ and therefore $[0,1]\cap\mathbb Q$$ is not closed.