The inverse of projection function is a closed map?

Question

I have this questione about the projection, be: $$\pi :X \times Y \to X$$ if we consider $ \pi^{-1}(x): x \to x \times Y $

I want to know if this is a closed map, is easy to see that $\pi$ is not, but I don't see what happens with the inverse.

Thanks.

Answer 1

$\pi$ is continuous. Thus its inverse image of an open set is open and a closed set closed. No, the inverse is not an open nor a closed map because it is not a map from X to X×Y.

It is a function from the power set of X to the power set of X×Y and those two power sets do not have a topology. Thus it is meaningless to ask if its open or closed.