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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?

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  • $\begingroup$ 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
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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.

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