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Chapter: Topological Aspects

We have already seen that the absolute value induces a metric space on the complex numbers. This fact allows applying many topological concepts also for the complex plane. We restate these concepts here for the special case of complex numbers.

Table of Contents

  1. Definition: Open and Closed Discs
  2. Definition: Closed and Open Regions of the Complex Plane
  3. Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
  4. Proposition: The distance of complex numbers makes complex numbers a metric space.
  5. Proposition: Complex Convergent Sequences are Bounded
  6. Definition: Continuous Functions at Single Complex Numbers
  7. Proposition: Convergent Complex Sequences Are Cauchy Sequences
  8. Definition: Complex Sequence
  9. Definition: Bounded Complex Sequences
  10. Definition: Convergent Complex Sequence
  11. Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences
  12. Definition: Limits of Complex Functions
  13. Definition: Complex Cauchy Sequence
  14. Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences
  15. Theorem: Completeness Principle for Complex Numbers
  16. Definition: Accumulation Points (Complex Numbers)
  17. Section: Paths
  18. Section: Homology and Winding Numbers
  19. Section: Homotopy
  20. Definition: Bounded Complex Sets

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References

Bibliography

  1. Riesz F., Sz.-Nagy, B.: "Functional Analysis (Translation)", Dover Publications, 1955
  2. Kneser, Hellmuth: "Mathematische Lehrbücher - Funktionentheorie", Vanderhoeck & Ruprecht, 1958
  3. Lang, Serge: "Complex Analysis", Springer, 1999, Forth Edition