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Chapter: Compact Sets

This chapter introduces the important concept of compact sets. Compact sets defined in any metric (or topological) spaces are a generalizations of closed real intervals known from the real analysis of one variable. The study of compact sets allows a deeper understanding of theorems known from the analysis of continuous functions with one real variable on bounded and closed intervals. The following table compares the results about compact sets with some basic results known from the real analysis of one variable:

Table of Contents

1. Definition: Heine-Borel Property Defines Compact Subsets
2. Proposition: Image of a Compact Set Under a Continuous Function
3. Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space
4. Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space
5. Proposition: Closed n-Dimensional Cuboids Are Compact
6. Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed
7. Proposition: Convergent Sequences are Bounded
8. Theorem: Heine-Borel Theorem
9. Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum
10. Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains
11. Proposition: Closed Subsets of Compact Sets are Compact

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References

Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984