What subbase generates metric topology?

Question

Let metric topology be the topology generated by metric balls of a metrizable space $X$

Is there a subbase $S$ that generates the metric topology?

I am asking because in most textbooks, it seems that people only discuss the base generated by the metric balls, and no attention is paid to subbase and no definition is given.

Thanks

Answer 1

A possible sub-base would be to take balls of only rational radius. Any ball of irrational radius can be realized as the infinite union of balls of rational radius.

E.g. $ \ \ B_\pi(x) = B_3(x) \cup B_{3.1}(x) \cup B_{3.14}(x) \cup \cdots$

I hope I've interpreted the definitions correctly. I'm taking "sub-base" to mean a subset $S \subset B$ that serves as a basis for the metric topology, where $B$ is the basis for the metric topology consisting of all open balls.