I am trying to understand the proof of Heine-Borel theorem given here.
If I understand the meaning behind the phrases in box, I think it is enough for me to understand the proof.
Please explain the meaning behind the phrases in the box.
2026-03-29 20:50:09.1774817409
Help me in understanding Heine-Borel theorem's proof
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open covering of $K$ = a collection of open sets such that $K$ is contained in the union of this collection
covers $K$ = $K$ is contained in their union
finite sub cover = there exist sets in this collection, finite in number, such that they also cover $K$
$B_j$ cover $K$ = the union of $B_j$ contains $K$.
As mentioned in the comments: A set $K$ is compact (in the general, topological way) if and only if for every cover (see definition above) $(U_\alpha)_{\alpha\in A}$ of $K$ that consists of open sets we can find a finite subcover $(U_{\alpha_j})_{j=1}^N$.