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I read that a map $f:X \rightarrow Y$ is continuous, if for any open set $O \subseteq Y$, the preimage $f^{-1}(O)$ is open in $X$. Alternatively, if $O$ is closed in $Y$, the preimage is closed in $X$.

However, suppose that $Y$ is equipped with the discrete topology where all subsets of $Y$ are clopen. Is there an instance where a continuous function maps an open set in $Y$ to an open set in $X$, but does not do so for closed sets ?

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If $f:X\to Y$ is continuous, then $f^{-1}[U]$ is open in $X$ for every open $U\subseteq Y$, and $f^{-1}[F]$ is closed in $X$ for every closed $F\subseteq Y$. It doesn’t matter what the topologies are: all that matters is that $f$ is continuous.

If $Y$ has the discrete topology, however, every subset of $Y$ is clopen, so $f^{-1}[A]$ is clopen in $X$ for every $A\subseteq Y$.

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