I need to determine whether the following set has accumulation points:
$0 \le \arg z<\pi/2 (z\ne 0)$
Would the accumulation point be z=0, as the set does not include 0? If not, does it not have any accumulation points as the set fans out to $\infty$ in the complex plane?
I think that I am just having a bit of trouble understanding the concept of an accumulation point.
Thank you for your help in advance!
Best Answer
An accumulation point of a set $A$ does not have to belong to $A$ itself. At least every interior point of $A$ and every non-isolated boundary point of $A$ is an accumulation point.
In your example, the set of accumulation points is the same as the closure of your set, i.e. the entire closed first quadrant $\{ z = x+iy | x \ge 0, y \ge 0 \}$. (In general, the set of accumulation points can be smaller than the closure: Take $A = \{ 0 \}$; then the set of accumulation points is empty.)