Srong topology, but really a question on covering spaces

[Office_Shredder]

Homework Statement

Prove that the set of maps {f in C_s¹(M,N)|f is a covering space} is open in the strong topology.

Homework Equations

The strong topology has as base neighborhoods sets of functions that are disjointly uniformly near f, along with their derivative, on compact subsets of a covering set of M

The Attempt at a Solution

So, I'm mostly having trouble with the covering space part. I understand how the strong topology works, and how proofs for things like the set of immersions, or the set of closed immersions, or the set of submersions is open work. But I understand those objects, and covering spaces are something I had to look up online.

From what I understand, around each point x in N I have an open subset of N, U, such that f^-1(U) is a bunch of disjoint sets each homeomorphic to U. I tried taking all of those as my cover of N, but then neither my cover of N nor M is locally finite and I wasn't able to figure out how to actually deduce anything. Then I tried saying take a locally finite chart of N {V_i}, then pick a point x in N. Around that point there is a U as above, and f^-1(U) is disjoint sets homeomorphic to U. Intersecting U with all the V_i's that x lie in give something homeomorphic to a subset of Rⁿ for some n (by the definition of a chart), and looking at f^-1(U intersect with those V_i) is disjoint subsets in M that must be homeomorphic to that same open subset.

Then if I pick a chart of M of {U_i such that f(U_i is a subset of V_i and K_i compact subsets that cover M, I can pick my e_i small enough so that |f-g|<e_i on K_i means that g(K_i) is a subset of V_i also. So my goal now is to find an open set around x that I can pull back to M via the inverse of g and get a bunch of homeomorphic copies of it. But I'm not really sure how to proceed at this point, since I don't know techniques to show that g is in fact a covering space, or what the criterion for it is besides the definition.

TL;DR I need to know what is required to make something a covering space/what techinques are good to show something is a covering space

 

[kurt.physics]

/what open set I can pull back to M via the inverse of g that will give me a bunch of homeomorphic copies.