# Finding the accumulation point

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!

• There are several accumulation points in this example. It's true that $z=0$ is one of the accumulation points, but the reason is not that $0$ is not in the set. What is your definition of an accumulation point? Jan 31, 2013 at 21:57
• Any point you can 'approach' from within the set is an accumulation point. Since $\frac{1}{n}$ is in the set and converges to $0$, then $0$ is an accumulation point. Jan 31, 2013 at 22:00
• The definition I have for an accumulation point is: point $z_0$is an accumulation point of S if each deleted neighborhood of $z_0$ contains at least one point of S. i.e. $\forall \epsilon >0 (D(z_0,\epsilon)\setminus {z_0})\cap S \neq \emptyset$
– Jess
Jan 31, 2013 at 23:58
• Another question I have regarding accumulation points: can they be within the set themselves?
– Jess
Feb 1, 2013 at 0:01

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.)
Since $$\operatorname{arg}(z) \geq 0$$, it includes the $$x$$-axis on the complex plane (except $$z=0$$). By contrast, $$\operatorname{arg}(z) < \frac{\pi}{2}$$, so the $$y$$-axis on the complex plane is excluded.
Hence, the set of accumulation points is $$\{ z = x + iy\, |\, x = 0, y \geq 0 \}$$.