I'm trying to solve this question from my textbook:

Let $f:X\rightarrow Y$ be a continuous map and let $Z \subset X$. Prove the inclusion $f(\bar Z)\subset\overline {f(Z)}$.

Thanks in advance for any help!


marked as duplicate by Tomás, Martin Sleziak, Jonas Meyer, Claude Leibovici, Micah Oct 30 '14 at 3:16

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  • 2
    $\begingroup$ Hint: let $y$ in $f(\overline Z)$ then $y=f(x)$ for some $x$ in $\overline Z$. By definition of $\overline Z$ this means that... $\endgroup$ – Did Apr 5 '14 at 9:30
  • $\begingroup$ I think the marked duplicate is for the converse, not what is being asked here. $\endgroup$ – ahorn Apr 8 '16 at 19:39

$Z\subset f^{-1}\left(\overline{f\left(Z\right)}\right)$ and as preimage of a closed set $f^{-1}\left(\overline{f\left(Z\right)}\right)$ will be closed because $f$ is continuous. Then $\bar{Z}\subset f^{-1}\left(\overline{f\left(Z\right)}\right)$ so $f\left(\bar{Z}\right)\subset\overline{f\left(Z\right)}$.


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