I have this theorem.
If is a metric space and . is closed in iff for some closed set in .
(I have proved a very similar theorem for open sets.)
However I cant seem to make any progress for the above theorem. (In either direction)
Going forward, we have is closed.
So is open in .
By using similar theorem for open sets.
for some open set in .
But trying to get back to by taking complements, I get
Which doesn't seem to help me at all.
Also using has all its limit points doesn't give me a lot to work with either.
This leads me to believe that the theorem is not true.
Is there something I am not considering? Even going backwards, I can't figure it out.
Thank you for your help.