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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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.