# Criterion for closed

• Sep 21st 2011, 09:09 AM
Sheld
Criterion for closed
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.

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 $\displaystyle X$.
The closure of $\displaystyle Y$, $\displaystyle \overline{Y}~,$ in $\displaystyle X$ is the 'smallest' closed set containing $\displaystyle Y$.
If $\displaystyle D\subseteq Y$ then $\displaystyle \overline{D}\subseteq\overline{Y}$.