# Equivalents metrics and closed sets

I have proven that if $d$ and $\rho$ are two equivalent metrics on a set $E$ then these metrics define the same open sets in both metric spaces $(E, d)$ as $(E, \rho )$. What I tried was that every open set in $(E, d)$ is also an open $(E, \rho)$ and each open set in $(E, \rho)$ is also an open $(E, d)$.

But I can not prove the above result for closed sets. How can I prove that each closed at $(E, d)$ is closed in $(E, \rho)$ and vice versa?

• If you've already proven that the open sets coincide, then it's easy. For a set is closed iff its complement is open. – Alex Provost Apr 6 '13 at 18:57

In other words: $U$ closed in $(E,d) \iff E-U$ open in $(E,d) \iff E-U$ open in $(E,\rho) \iff U$ closed in $(E,\rho)$.