Explanation: Explanation of the Heine-Borel Property

(related to Definition: Heine-Borel Property Defines Compact Subsets)

The definition of the Heine-Borel property causes sometimes great difficulties to students. This definition does not(!) state that

"$U$ is compact, if $U$ has a finite cover".

This interpretation is wrong, since \(U\subset X\) has always a finite cover, for instance the cover consisting of the open set \(X\) alone. The definition rather requires that any given open cover of \(U\) contains is a finite subcover.

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