Let $X$ be topological space, and $A \subseteq X$ that is bounded. Is the closure of $A$ also bounded?

This is true if $X$ is topological vector space, but is it if $X$ is only topological?

  • $\begingroup$ What is your definition of closure? $\endgroup$ – Ilham Apr 23 '15 at 20:48
  • $\begingroup$ And how to define boundedness without a metric $\endgroup$ – marwalix Apr 23 '15 at 20:48
  • $\begingroup$ Is this a metric space? $\endgroup$ – Rellek Apr 23 '15 at 20:49
  • $\begingroup$ Closure is well defined for a general topological space. $\endgroup$ – Sam Clearman Apr 23 '15 at 20:49
  • $\begingroup$ @Yeah, I just wanted to know what he meant by closure in his course. So we could work with his definition to avoid wasting time when writing a proof. But as it stands boundedness is a metric property. $\endgroup$ – Ilham Apr 23 '15 at 20:50

If $X$ is only topological, "bounded" has no meaning.

If $X$ is a metric space, the answer is "yes", as the closure consists of points that are within $\epsilon$ of points in the set.


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.