Open ball with rational radii forms a basis.

Question

Show that in a metric space, the set of all balls with rational radii is a basis for the topology.

Although I understand the question, I have no clue at all. I am very new to topology, can anyone give me some idea?

Answer 1

By definition, the collection of all open balls is a base for the topology.

Now let $B$ be an open ball with centre $a$, and irrational radius $\rho$. Then $B$ is the union of all the open balls with centre $a$ and rational radius $r\lt \rho$. Thus every open ball is a union of open balls with rational radius.

It follows that every open set is a union of open balls with rational radius. Thus these form a base for the topology.