Open sets in a metric space

The question is:

Let be a metric space, and be a subset of .

If is open, then is open in the metric space .

Pf:

implies is a metric space is open.

Therefore is closed

is open, thus is closed.

So is closed, since it is the finite union of closed sets.

Therefore is open.

I seem to follow all the definitions and theorems correct. Is this ok?

There is also a second part to this question.

Conversely, if is open in , an open set s.t

I don't get this question. Don't we already have an open set that works in the hypothesis? E = E_1?

Any help is greatly appreciated. Thank you.

-Jame