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, $\displaystyle \bigcup_{k \in N}\{k\}$ is an infinite union of compact sets, which is a set of natural numbers N.
Take an open interval, let's say, $\displaystyle I_x=(x-1/2, x+1/2), x \in N$. Then $\displaystyle C=\{I_x:x \in N\}$ 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.