Question

I was reading this question: http://mathoverflow.net/questions/25578/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?

Answer 1

Any uniformly continuous map $f$:X→Y between metric spaces induces a uniformly continuous map $C\mapsto \overline{f(C)}\ $ between the spaces of closed subsets wrto the Hausdorff distances; in fact with the same modulus of continuity. (Just recall that the Hausdorff distance between A and B is less than δ if and only if for any a∈A there is some b∈B with d(a,b)<δ and for any b∈B, there is some a∈A with d(a,b)<δ).

PS: As to the topologic side of the question (stability of topological or homotopical properties of subsets of a space under perturbations in Hausdorff distance). As far as I know these things usually work well in an ANR metric space X, because of the homotopy extension property. For instance, any closed subset of X that is contractible in X (meaning that the inclusion map $C\to X$ is null homotopic) has a contractible nbd in X. If X is also compact, the nbd is a uniform nbd, so if a sequence of closed sets $C_n$ converges in the Hausdorff sense to C, the $C_n$ are eventually contractible in X, too. Generalizing a bit, we may also say that the Lusternik-Schnirelman category of C (minimum cardinality of a contractible closed covering of C) is lower semicontinuous wrt the Hausdorff distance (the category of the limit is larger or equal to the limit of the category along the sequence). Reference (for ANR's and homotopy): e.g. the first 2 chapters of Spanier book.