Given two convex set below:
$C = \{ x \in \mathbb{R}^2 | x_2 \leq 0\}$
and
$D = \{ x \in \mathbb{R}_+^2 | x_1 x_2 \geq 1\}$
Which one is closed/open and how that is determined? My idea about a closed set is that it will be closed when its bounded (like a circle), but here both of them seems to unbounded (?). Would be grateful if someone explains the concept in detail.
Closedness is connected with limits of sequences. Namely, $A$ is a closed set (in a metric space), if for any sequence $(x_n)$ of elements of $A$ the following condition holds: $$\lim\limits_{n\to\infty}x_n=x\implies x\in A.$$
Both $C$ and $D$ are closed. Try to check the above condition.
In $\Bbb R^n$ the sets which are both closed and bounded are compact. This is considerably more than closedness.