2
$\begingroup$

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?

$\endgroup$
  • 3
    $\begingroup$ 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
1
$\begingroup$

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 $(E,\rho)$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.