Equivalents metrics and closed sets

Question

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 06 '13 at 18:57

score 1 · Accepted Answer · answered Apr 06 '13 at 19:12

1

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 $(E,\rho)$.

answered Apr 06 '13 at 19:12

Alex Provost

20,991

Thanks Silencer...I regret not having seen this test easier. Excuse me. – Roiner Segura Cubero Apr 06 '13 at 19:52
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. – Alex Provost Apr 06 '13 at 19:54

Equivalents metrics and closed sets

1 Answers1