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.