A map $f:X \rightarrow Y$ between two topological spaces $(X,\tau_X)$ and $(Y,\tau_Y)$ is said to be closed if for any closed subset $C\subseteq X$, the image $f(C)\subseteq Y$ is also closed in $Y$.
In mathematical notation, $f$ is a closed map if for all closed $C\subseteq X$, we have $f(C)\subseteq Y$ is closed:
$$ \forall C\subseteq X,\ \text{if}\ C\text{ is closed in}\ (X,\tau_X),\ \text{then}\ f(C)\text{ is closed in}\ (Y,\tau_Y). $$This means that the inverse image of any closed set in $Y$ under $f$ is also closed in $X$.
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Author of the notes: Antonio J. Pan-Collantes
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