(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.