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Let $X,Y$ be two topological spaces. Let $f:X\to Y$ be a map from $X$ to $Y$.

Choose the CORRECT statement from the list.

If for every closed set $B$ in $X$ the image $f(B)$ is a closed set in $Y$, then $f\,$ is a continuous map.

If for every closed set $B$ in $Y$ the inverse image $f^{-1}(B)$ is a closed set in $X$, then $f\,$ is a continuous map.

If for every open set $A$ in $X$ the image $f(A)$ is an open set in $Y$, then $f\,$ is a continuous map.

If for every set $A$ with empty boundary in $Y$ the inverse image $f^{-1}(A)$ is a set with empty boundary in $X$, then $f\,$ is a continuous map.