Showing a set is open but not closed Let $X = \{ (x,y) \in \mathbb{R}^2 : x > 0, y \geq 0 \} $. I am claiming $X$ is open, but it is not closed. 
My Try:
To show it is not closed, I found a sequence that converges to a point outside $X$. For instance, $(x_n, y_n) = (\frac{1}{n}, \frac{1}{n} ) \to (0,0) \notin X$. Hence, $X$ cannot be closed. 
In trying to show it is open, I have diffictulty trying to find a suitable open ball contained in $X$. Any help would be greatly appreaciated. 
Thanks.

The set is not open. Take for example the point $P=(5,0)$. There is no ball with centre $P$ which is fully contained in our set.  
