I have the following set: $B = \left\{(x,y)\in\mathbb{R^2}: 0<x\leq1, y\geq2\right\}$
I know that that a set is open if for every point in that set there is a number $\epsilon$ such that a ball of radius $\epsilon$ around that point is a subset of the set, and closed if it contains its limit points. Looking at this set, I can find examples of limit points it contains, and limit points it doesn't contain. For example, the set contains the point $(1,2)$ but doesn't contain the point $(0,2)$. So I want to say it's neither closed nor open. But I'm fairly sure this is incorrect. What am I missing here?
Best Answer
On the contrary, your reasoning is entirely correct, although I believe there is some mistake on your definition of open set, since open means that any point has a neighborhood contained on the set. Open and closed sets are not exhaustive in general, so you can have sets which are not open nor closed, a simpler example being $(0,1]$ in the usual topology of $\mathbb{R}$.