MHB Complex accumulation points and open/closed

[Dustinsfl]

All complex numbers of the form $(1/n) + (i/m)$, $n,m\in\mathbb{Z}^+$.Complex numbers aren't well ordered so how is this treated?

 

[Sudharaka]

All complex numbers of the form $(1/n) + (i/m)$, $n,m\in\mathbb{Z}^+$.Complex numbers aren't well ordered so how is this treated?

Hi dwsmith, :)

The notions of accumulation points, openness, closeness are defined for Topological spaces in general. Those definitions could be adapted to the set of real numbers or complex numbers. In Complex Variables and Applications by James Brown and Ruel Churchill you can find a good introduction about how these concepts are defined in the context of complex numbers. The approach is rather geometrical(you have to visualize the set in the Argand plane) but still I find it very intuitive.

All the points, \(\frac{1}{n}\mbox{ where }n\in\mathbb{Z}^+\) as well as \(\frac{i}{m}\mbox{ where }m\in\mathbb{Z}^+\) are limit points. Additionally zero is also a limit point. This set is neither open nor closed.

Herewith I have attached the relevant pages of Complex Variables and Applications by James Brown and Ruel Churchill for your reference.

Kind Regards,
Sudharaka.