I was reading this question: limiting behaviour of converging loops on a torus

And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their projections must converge in your $S^1$", but a quick google search gives me no results along these lines- do they exist? If not why not?

I am aware that if either of your spaces are unbounded then a sensible topology isn't particularly forthcoming, but is there a situation in which a result of this form can make sense? As a starting point let's set the bar at:

Do compact fibrations induce maps on their subsets that are continuous wrt Hausdorff distance?

Can we do better? Can we do a little worse? Or does none of this make sense?