What is the intuition of covering spaces?
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I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics.
The definition of covering space is as follows:
Let XX be a topological space. A covering space of XX is a topological space CC together with a continuous surjective map p:C→Xp:C→X such that for each x∈Xx∈X there is an open neighborhood UU of xx such that p−1(U)p−1(U) is the disjoint union of open sets which are homeomorphic to UU through pp.
The definition means that for each x∈Xx∈X there's U⊂XU⊂X which is open with x∈Ux∈U which that there are {Vα⊂C:α∈A}{Vα⊂C:α∈A}, with all of the VαVα open, satisfying Vα∩Vβ=∅Vα∩Vβ=∅ when α≠βα≠β and p|Vαp|Vα being one homeomorphism between VαVα and UU with
p−1(U)=⋃α∈AVα.
p−1(U)=⋃α∈AVα.
Now, although the definition is fine, I want to get some intuition about it.
When we define covering spaces, what is the intuitive thing which we are really turning rigorous with a precise definition? What is really the idea behind this definition?
I really couldn't get a nice intuition regarding this. Also, what is the importance of this definition, in the sense of when do we expect to see this concept becoming useful?
The definition of covering space is as follows:
Let XX be a topological space. A covering space of XX is a topological space CC together with a continuous surjective map p:C→Xp:C→X such that for each x∈Xx∈X there is an open neighborhood UU of xx such that p−1(U)p−1(U) is the disjoint union of open sets which are homeomorphic to UU through pp.
The definition means that for each x∈Xx∈X there's U⊂XU⊂X which is open with x∈Ux∈U which that there are {Vα⊂C:α∈A}{Vα⊂C:α∈A}, with all of the VαVα open, satisfying Vα∩Vβ=∅Vα∩Vβ=∅ when α≠βα≠β and p|Vαp|Vα being one homeomorphism between VαVα and UU with
p−1(U)=⋃α∈AVα.
p−1(U)=⋃α∈AVα.
Now, although the definition is fine, I want to get some intuition about it.
When we define covering spaces, what is the intuitive thing which we are really turning rigorous with a precise definition? What is really the idea behind this definition?
I really couldn't get a nice intuition regarding this. Also, what is the importance of this definition, in the sense of when do we expect to see this concept becoming useful?
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