So I did this:Prove that a subset of a metric space is open iff it is a union of open balls.

Suppose a subset of a metric space is open.

Let be a metric space.

Let

Then there exists some such that .

But is an arbitrary element of .

Therefore centre a ball around every point in .

.

(Since the centre of each of these balls is a point in , the union of them must be the entire set .)

Conversely, suppose a subset of a metric space is .

Then by definition is open.

Is this right?