Find an infinite collection {Sn : n is an element of N} of compact sets R such that the union from n=1 to infinity Sn is compact.
Each singleton set is compact in the standard topology on R.
Take an infinite union of singleton sets, let's say, is an infinite union of compact sets, which is a set of natural numbers N.
Take an open interval, let's say, . Then is an open cover for N, but it has no finite subcover.
You can find other examples in other topological spaces, such as a discrete topology on N.