TITLE: How can I tell whether this set is closed or open?
QUESTION [7 upvotes]: the question I'm having trouble with is this:
$$A:=\bigcup_{n=1}^{\infty}\left({\left[\frac{1}{n+1},\frac{1}{n}\right) \times \left(0,n\right)}\right).$$
In the Euclidean space $\mathbb{R}^2$, is the subset $A$ either closed or open?
I drew a picture in $\mathbb{R}^2$ and I really think it's open, but I can't come up with an elaborate way to describe why. Any help would be appreciated.
Thanks.

REPLY [4 votes]: Pick a point in the set. It must belong to one of the products in the union.
Say, it belongs to $\left[\frac{1}{k+1},\frac{1}{k}\right[ \times \left]0,k\right[$.
Let the point be $(x,y)$. We have $\frac1{k+1} \le x < \frac1k$ and $0 < y < k$.
Now, we consider $\left]\frac1{k+2},\frac{x+1/k}2\right[ \times \left]\frac y2,\frac{y+k}2\right[$ and prove that it is contained in $A$.
The set is also open
Since there is an open set around every point in $A$, we can conclude that $A$ is open.