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.
 A: When we have an interval $I=(a,b)$ or $I=(a,\infty)$, you have to realize that $a\notin I$ is not part of the interval.
In fact $x\in I$ means $x>a$, therefore the quantity $r=\frac{x-a}2>0$ and you have a ball centered in the midpoint $B(\frac{x+a}2,r)\subset I$.
This characterization works well for intervals because they are convex (notice the demo uses midpoint), but it's not much different for general open sets in $\mathbb R^n$.
See a closed set as a blob with a skin, the skin being the border (i.e. $\bar A-\mathring A$, the closure minus the interior), and an open set as a skinless fluffy blob. A point in the open set can be arbitrary close to the border but there is always space to between the point and the "would be" border, this is what the included ball characterization says. The closer to the "would be" border you are, the smaller the ball.
Notice that a set could be nor open nor closed, visualize this as a hole or a scratch in the skin (i.e. some limits points are attained but not all). In $\mathbb R$, the interval $(a,b]$ is such a set, open in $a$ but closed in $b$, and nor closed nor open as a whole.
A: 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 $\bigl(y-(y-x),y+(y-x)\bigr)$.
