compact sets

We will again give the definition of the compactness.

If a open cover of $X$  always contains a finite sub cover, the set $X$  is compact.

The reasons why the compactness is more stronger than the completeness depend
on two propositions as follow.

(1)A compact space becomes complete.
(2)A compact set becomes bounded.

We leave out the proofs. However, as a open cover of $X$  always contains a finite sub cover,
no one will have a feeling of strangeness.

Therefore, a set becomes compact if and only if a set is complete and has a finite open cover.
Then we reach the theorem of Heine Borel because a complete subset in a complete space is a closed set. (a finite open cover means bounded.)