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Topology of complex numbers

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Assume S is a subset of the complex number set

1. an interior point of S
2. an exterior point of S
3. a boundary point of S
4. an accumulation point of S

Please make the explanations clear and give examples.

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Solution Summary

The solution discusses the topology of complex numbers. Interior, exterior, boundary points and accumulation points are analyzed.

Solution Preview

Please see the attached file.

The set S in the complex plane define some Euclidean area.
We first define the concept of neighborhood. A neighborhood of a point z is the set that contains all points such that

In other words, the neighborhood of z is a set that contains all the points inside a circle centered about z and has a radius

z is an interior point of the set if there exists such that
An interior point of S is a point which is included in S and all the points around it within an infinitesimally small radius are also members of the set S.
All the interior points of S form a set called the interior of S and is denoted by
 
Example.
The point is an interior point of the set which is the right half complex plane. If we choose then for any we define

So:

And

Therefore
So we found such that any point that satisfies is in the set S, therefore is an interior point ...

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