Let's suppose I have a topological space $X$, for which I am trying to prove its compactness. If I construct a function $f: X \to Y$ to another topological space $Y$, what are some examples of conditions on $f$ and $Y$ that could guarantee that $X$ is compact?
E.g. if $f$ is open and injective, and $f(X)$ is compact, then $X$ is compact [1]. One could prove the compactness of $f(X)$ by showing $Y$ is compact and $f(X) \subseteq Y$ is closed.
Does assuming that either $X$ or $Y$ are Hausdorff / metrisable help?
[1] This seems dual to the theorem that continuous images of compact spaces are compact.
The other way round is easier: if we can have a continuous $f: Y \to X$ that is onto and continuous and $Y$ is compact then so is $X$. E.g. that's why the image of a path is always compact.
For your case $f: X \to Y$ we could use $f$ perfect: continuous, closed, onto and all fibres compact (so all sets of the form $f^{-1}[\{y\}]$ are compact); then $Y$ compact implies $X$ is too. This can be used to see that $X \times Y$ is compact whenever $X$ and $Y$ are (using the projection maps).