Which of the following statements are true?
(a) Every connected subset of $\mathbb{R^n}$ is convex
(b) Every convex subset of $\mathbb{R^n}$ is compact
(c) Every convex subset of $\mathbb{R^n}$ is connected
(d) The set $\left\{ (x,y) \in \mathbb{R}^2 : x \geq 0, y \geq 0 \right\} \cup \left\{ (x,y) \in \mathbb{R}^2 : x \leq 0, y \leq 0 \right\}$ is convex
It is given that option $c$ is correct. I agree if the set is convex then clearly it is connected too. And connected need not be compact and convex set also need not be compact so option (a) and (b) are false. But, what about option $d$? I think it should also be correct, as the union is the entire $\mathbb{R}^2$ plane, it must be convex. Am I correct?
The set described in d) is not $\mathbb{R}^2$, it is the union of 1st and 3rd quadrants of the cartesian plane. It is connected because $0$ belongs to the set but it is clearly not convex