My text book said:
Not every metrizable space is locally compact.
And it lists a counterexample as following: The subspace $Q=\{r: r=\frac pq; p,q \in Z\}$ of $R$ with usual topology, i.e., $Q$ is the set of all rationals. It said: for any open ball of any point $r \in Q$, the closure is not compact. I can't understand this sentence. Why the closures of the open balls are not compact.
Could anybody help me to understand this sentence. Thanks ahead:)