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6575Definition: 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 \(U\) (and is due to Heinrich Eduard Heine and Émile Borel.

1 The above definition makes no reference to any kind of a metric of the space $X$. Thus, it can be used in more generalized topological spaces rather than metric spaces.

| | | | | Contributors: bookofproofs | References: [582]

65761.Explanation of the Heine-Borel Property


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Bibliography (further reading)

[582] Forster Otto: “Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen”, Vieweg Studium, 1984

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