product of topological spaces

Question

I was just wondering, if $X$ is a topological space and $A$ is a closed subset of the topological space $Y,$ where $f$ is the function mapping $f:X\rightarrow Y$, then is the product $X\times A$ also a closed subset of $X\times Y?$ (Note $Y$ here is compact and Hausdorff)

Quite frankly, in the notes it says it is closed but I do not know why it is the case, can anyone help me explain why it is the case?

Answer 1

Your question should be: is $X \times f(A) $ close subset of $X \times Y $? And the answer is no, $f$ has to be continuous and open or close mapping.