Srong topology, but really a question on covering spaces

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In summary, the problem is to prove that the set of maps that are covering spaces is open in the strong topology. This topology has base neighborhoods that are sets of functions that are uniformly near f, along with their derivative, on compact subsets of a covering set of M. The difficulty lies in understanding covering spaces, and the attempted solution involves picking a locally finite chart of N {Vi} and a chart of M {Ui} such that f(Ui) is a subset of Vi. The goal is to find an open set around x that can be pulled back to M via the inverse of g and give a set of homeomorphic copies. The techniques for showing something is a covering space and the criteria for it are not known.
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Homework Statement



Prove that the set of maps {f in Cs1(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 {Vi}, 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 Vi'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 Vi) is disjoint subsets in M that must be homeomorphic to that same open subset.

Then if I pick a chart of M of {Ui such that f(Ui is a subset of Vi and Ki compact subsets that cover M, I can pick my ei small enough so that |f-g|<ei on Ki means that g(Ki) is a subset of Vi 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
 
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/what open set I can pull back to M via the inverse of g that will give me a bunch of homeomorphic copies.
 

What is a covering space?

A covering space is a topological space that maps onto another space, called the base space, in a continuous and surjective manner.

How is a covering space different from a quotient space?

A quotient space is formed by collapsing a space onto itself through an equivalence relation, while a covering space maps onto the base space in a one-to-one and onto manner.

What is the significance of the universal covering space?

The universal covering space is the simply connected covering space of a given space. It provides a way to understand the topology of a space by studying its covering spaces.

Can a space have multiple covering spaces?

Yes, a space can have multiple covering spaces. In fact, the same space can have infinitely many different covering spaces.

What is the relationship between covering spaces and fundamental groups?

The fundamental group of a space is closely related to its covering spaces. In fact, the fundamental group can be used to classify covering spaces of a given space.

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