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How do I find the example of a continuous surjective function $f: X \rightarrow Y$ such that inverse image of dense set is not dense?

Can anyone suggest some hints for this question?

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    $\begingroup$ Try to find a proof that the inverse image of a dense subset under a continuous function is dense. See why such a proof might fail. This should help you find a counterexample. $\endgroup$ – Stephen Montgomery-Smith Sep 5 '20 at 16:25
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Hint: what if $X$ has the discrete topology?

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What about

$$\begin{array}{l|rcl} f : & X= \mathbb R & \longrightarrow & Y = [0,\infty) \\ & x & \longmapsto & \begin{cases}0 & x <0\\ x & x \ge 0\end{cases}\end{array}$$

$(0,\infty)$ is dense in $Y$ but $f^{-1}[(0,\infty)] = (0,\infty)$ is not dense in $\mathbb R$.

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