Understanding open and closed sets I am really struggling with understanding the notions of open and closed sets.
I "understand" that an open set is one which contains a ball with some radius $r$ for each $x$ in that set such that the ball is still inside the set. A closed set contains all of its limit points.
But given the set $$(-\frac{1}{n},\infty),$$ I am unable to "picture" in my head why this is open. If anyone could please give advice on how to look at open and closed sets because so far most definitions have not made it any easier. Thank you.

Forget the $-\frac1n$. For any real number $x$, $(x,\infty)$ is an open set. That's so because, if $y\in(x,\infty)$, then $(x,2y-x)\subset(x,\infty)$. And $(x,2y-x)$ is the open ball centered at $y$ with radius $y-x$, since it is equal to $\bigl(y-(y-x),y+(y-x)\bigr)$.
