I'm still confused about this. There must be something that I've missed.
2.34 Compact subsets of metric spaces are closed.
2.32 $K$ is compact if every open cover of $K$ contains a finite subcover.
Now suppose $K$ is open in $X$. Then we get $K \subseteq K, K \subseteq \bigcup K$ (obviously).
But then $K$ is finitely covered by a series open sets (actually just one), so it is compact and yet we know that it is open, not closed.