Is this subset of $\mathbb{R}^2$ closed? I have the set $S = \{(x,y) | x>0, y\geq0\}$ and am asked if it is open/closed. I have proven that it's not open but I'm confused about whether it's closed, as its complement is not open.
Am I right in saying it's closed because points of the form $(x,0)$ are limit points but not in $S$? 

It is neither open nor closed.
Open means all points are interior but the points $(x,0)$ for $x>0$ are not interior ($\forall r\in \mathbb R_+$ there is a point in $N_r((x,0))$ which $\not\in S$)
Closed means it contains all its limit points but points of the form $(0,y)$ for $y\geq0$ are limit points (i.e. $\forall r \in \mathbb R_+$, $N_r((0,y))$ contains a point of $S$) but these points are not in the set $S$ since $x>0$.
