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Definition: Heine-Borel Property Defines Compact Subsets

A subset \(U\) of a metric (or topological) space¹ \(X\) is called compact, if for every open cover $(U_i)_{i\in I}$ of $U$ there exist only finitely many indices \(i_1,i_2,\ldots,i_k\in I\) with

\[U\subset U_{i_1}\cup U_{i_2}\cup \ldots \cup U_{i_n}\].

Notes

• The finite union $U_{i_1}\cup U_{i_2}\cup \ldots \cup U_{i_n}$ is sometimes referred to as an open subcover.
• The existence of a finite subcover for every open cover of $U$ is called the Heine-Borel property of \(U\), called after o.

Table of Contents

Explanations: 1

Mentioned in:

Chapters: 1
Corollaries: 2 3
Explanations: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Propositions: 20 21 22 23 24 25 26
Theorems: 27 28 29 30

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References

Bibliography

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

Footnotes