- #1
Dustinsfl
- 2,281
- 5
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?
Complex accumulation points and open/closed
In summary, the conversation discusses the treatment of complex numbers of the form $(1/n) + (i/m)$, $n,m\in\mathbb{Z}^+$, and their relation to concepts such as accumulation points, openness, and closeness in Topological spaces. The book Complex Variables and Applications by James Brown and Ruel Churchill is recommended as a resource for a geometrical understanding of these concepts in the context of complex numbers. It is noted that all points of the form $\frac{1}{n}$ and $\frac{i}{m}$ are limit points, and that the set is neither open nor closed. Attached pages from the book are also provided for further reference.
- #2
Sudharaka
Gold Member
MHB
- 1,568
- 1
dwsmith said:
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.
1. What are complex accumulation points?
Complex accumulation points, also known as limit points, are points in a complex set that can be approached infinitely closely by an infinite number of other points in the set. They can be both real and imaginary numbers and are used to define open and closed sets in complex analysis.
2. How are complex accumulation points different from ordinary accumulation points?
Unlike ordinary accumulation points, which are defined in terms of epsilon neighborhoods, complex accumulation points are defined in terms of circular neighborhoods. This means that they take into account both magnitude and direction in the complex plane.
3. What is an open set in complex analysis?
An open set in complex analysis is a set that contains all its complex accumulation points. This means that for every point in the set, there exists a circular neighborhood around that point that is fully contained within the set. In other words, the set does not include its boundary points.
4. What is a closed set in complex analysis?
A closed set in complex analysis is a set that includes all its complex accumulation points. This means that for every point in the set, there exists a circular neighborhood around that point that contains at least one point in the set. In other words, the set includes its boundary points.
5. How are open and closed sets related to complex accumulation points?
Open and closed sets are defined in terms of complex accumulation points. An open set contains all its accumulation points, while a closed set includes all its accumulation points. This means that a set can be both open and closed, if it contains all its accumulation points and does not have any accumulation points outside of the set.
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