I see now, I didn't scroll down far enough. It has to be incorrect then, because the preimage of a closed set under a continuous map is always closed (this is even sometimes used as a definition for continuity).
Perhaps the person writing the answers was thinking about compact sets, because then the statement becomes false (there are compact sets
such that 
is not compact for the function 
in the question).
Perhaps the person writing the answers was thinking about compact sets, because then the statement becomes false (there are compact sets
such that is not compact for the function in the question).