What can we say about the product space (with product topology) $X\times Y$ if $X$ is an arbitrary topological space and $Y$ is a compact Hausdorff space?
Do we know from the given information if the projection $\pi:X\times Y\to X$ is open/closed?
Regards
The product topology is defined in such a way that the projection maps are open. A projection map parallel to a compact factor is always closed; the factor need not be Hausdorff. This is a standard basic result in topology; you can find a proof in the first paragraph of this answer.
Beyond that very little can be said about $X\times Y$.
