Prove that the set
$\left\{(x,y)\in \mathbb{R}^2|0<y\le 1\right\}$
is neither closed nor open.
My book only has definitions (no examples) and I understand them but have no idea how to transfer them onto actual problems, so I'm struggling here. Any help would be appreciated, thank you
Edit: Definitions:
A set $A$ is open if for all $\vec{x}$ $\epsilon$ $A$, there exists an $r > 0$ such that $B_r(\vec{x})$ $\subseteq A$ (where $B_r(\vec{x})$ is the open ball centered at $\vec{x}$ with radius $r$)
A set $A$ is closed if $A^c$ is open.
Best Answer
Consider $B_r(1)$. You should notice for any $r>0$, $B_r(1)$ contains both point in $(0,1]$ and not in $(0,1]$. This implies that neither $(0,1]$ or its complement are open.