The question is:
Let be a metric space, and be a subset of .
If is open, then is open in the metric space .
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.