How can a compact set on R^n be closed (heine borel), when by definition a compact set has to be finitely covered by open sets?

Question

A compact set A is defined st given a cover of open sets of A, there exist a finte subcover with elements in that original cover. And so A is defined as a finite union of open sets, and so A should be open (def of a topological space, the intersection of two opens is open itself). By heine borel, in R^n a compact set is closed (and bounded). The question is therefore: is A a closed or an open set?

Answer 1

The key word here is “covered”. A set $K$ is compact if, when $K$ is contained in an union of open sets, then $K$ is contained in the union of finitely many of those sets. It is not being claimed that $K$ is the union of those sets.