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?

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    $\begingroup$ If you've already proven that the open sets coincide, then it's easy. For a set is closed iff its complement is open. $\endgroup$ – 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 $(E,\rho)$.

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  • $\begingroup$ Thanks Silencer...I regret not having seen this test easier. Excuse me. $\endgroup$ – Roiner Segura Cubero Apr 6 '13 at 19:52
  • $\begingroup$ No problem! These kinds of "duality" arguments show up very often in mathematics. Sometimes, doing half of the problem is all that's really needed. $\endgroup$ – Alex Provost Apr 6 '13 at 19:54

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