Confirmation that These Sets are Open

Question

Are These Open Sets?

(i) $\mathbb{R}$ - $\mathbb{Q}$

(ii) $\mathbb{R}$ - {$0$}

(iii){$x \in \mathbb{R}$: $x^2>4$}

My thoughts were that (i) = yes, (ii) = No, and (iii) = yes

Answer 1

Recall that an open set is a set such that every point in the set has some ball around it that is completely contained in the set. Note that $\mathbb{R} - \mathbb{Q}$, no matter how small you take an interval around some point $x$, it will contain some rationals as the rationals are dense. So i) is not open. ii) is open; every point has an open neighbourhood in the set; simply take the interval around $x$ with radius $\frac{|x|}{2}$. iii) is also open; for positive $x$, take an interval around $x$ with radius $\frac{x-2}{2}$, and use a similar argument for negatives.