I have this theorem.

If $\displaystyle X$ is a metric space and $\displaystyle Y \subset X$. $\displaystyle D \subset Y$ is closed in $\displaystyle Y$ iff $\displaystyle D = C \cap Y$ for some closed set $\displaystyle C$ in $\displaystyle X$.

(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 $\displaystyle D$ is closed.

So $\displaystyle Y-D$ is open in $\displaystyle Y$.

By using similar theorem for open sets.

$\displaystyle Y-D = E \cap Y$ for some open set $\displaystyle E$ in $\displaystyle X$.

But trying to get back to $\displaystyle D$ by taking complements, I get

$\displaystyle D = X-E \cup X-Y$

Which doesn't seem to help me at all.

Also using $\displaystyle D$ 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.