description/proof of that surjective local homeomorphism (especially, covering map) is quotient
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of surjection.
- The reader knows a definition of local homeomorphism.
- The reader knows a definition of quotient map.
- The reader admits the proposition that any local homeomorphism is continuous.
- The reader admits the proposition that for any map between any sets, the composition of the map after the preimage of any subset of the codomain is identical if the map is surjective with respect to the argument subset.
- The reader admits the proposition that any local homeomorphism is open.
Target Context
- The reader will have a description and a proof of the proposition that any surjective local homeomorphism (especially, any covering map) is a quotient map.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(: T_1 \to T_2\), \(\in \{\text{ the surjective local homeomorphisms }\}\)
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Statements:
\(f \in \{\text{ the quotient maps }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(f\) is continuous; Step 2: take any subset, \(S_2 \subseteq T_2\), such that \(f^{-1} (S_2) \subseteq T_1\) is open; Step 3: see that \(S_2 = f \circ f^{-1} (S_2)\); Step 4: see that \(f\) is open; Step 5: conclude the proposition.
Step 1:
\(f\) is continuous, by the proposition that any local homeomorphism is continuous.
Step 2:
Let \(S_2 \subseteq T_2\) be any subset such that \(f^{-1} (S_2) \subseteq T_1\) is open.
Step 3:
\(S_2 = f \circ f^{-1} (S_2)\), by the proposition that for any map between any sets, the composition of the map after the preimage of any subset of the codomain is identical if the map is surjective with respect to the argument subset: \(f\) is surjective.
Step 4:
\(f\) is open, by the proposition that any local homeomorphism is open.
Step 5:
By Step 3 and Step 4, \(S_2\) is open on \(T_2\).
So, \(f\) is a quotient map.
