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Topology of complex numbers

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.

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 ...

Solution Summary

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

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