# Proving that $f(\bar Z)\subset\overline {f(Z)}$ when $f$ is a continuous map [duplicate]

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, MicahOct 30 '14 at 3:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 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... – Did Apr 5 '14 at 9:30
• I think the marked duplicate is for the converse, not what is being asked here. – ahorn Apr 8 '16 at 19:39

## 1 Answer

$Z\subset f^{-1}\left(\overline{f\left(Z\right)}\right)$ 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 $f\left(\bar{Z}\right)\subset\overline{f\left(Z\right)}$.