I just learned compactness today in college analysis course (we are using baby Rudin for textbook). The Rudin's definition is
Open cover: By an open cover of a set $E$, in a metric space $X$, we mean a collection $\{G_a\}$ of open subsets of $X$ such that $E \subset \cup_{\alpha}G_{\alpha}$
Compactness: A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover.
Professor said that open set like (0, 1) is not compact, but I just thought that why can't we just use (0, 1) itself as a finite subcover because (0, 1) is just one, which I mean here as "finite," set which actually covers (0,1).
I would greatly appreciate it if you could enlighten me so that I can truly understand compactness.
Thank you!