Please give examples and how to figure them out

Question

Give examples of (i) a set that is not a closed, bounded interval, but nevertheless contains its supremum and infimum, and (ii) a set that is not an open interval but nevertheless does not contain it supremum or infimum. I've tried to look into the definition of open sets and closed sets. There's a question which I've proved earlier Q. Show that a closed, bounded interval contains is supremum and infimum and that an open interval contains neither. But I couldn't think of an example.

This is a question from Scharamm's book of Real Analysis. Edit: I realised that not open sets don't mean closed sets. P.S I'm a beginner in Analysis. Also I am new to stackexchange community.

Answer 1

i. Remove a point from a closed bounded interval.

ii. A half open set.

Answer 2

The Cantor set is not a closed bounded interval, yet contains its minimum (infimum) $0$ and maximum (supremum) $1$.

A set like $\{x \in \mathbb{Q}: x^2 < 2\} \subseteq \mathbb{R}$ is not an open interval and does not contain its infimum $-\sqrt{2}$ and its supremum $\sqrt{2}$.