description/proof of that open continuous bijection is homeomorphism
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of open map.
- The reader knows a definition of bijection.
- The reader knows a definition of homeomorphism.
Target Context
- The reader will have a description and a proof of the proposition that any open continuous bijection is a homeomorphism.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(: T_1 \to T_2\), \(\in\{\text{ the open maps }\} \cap \{\text{ the continuous maps }\} \cap \{\text{ the bijections }\}\)
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Statements:
\(f \in \{\text{ the homeomorphisms }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(f^{-1}\) is continuous.
Step 1:
As \(f\) is a bijection, there is the inverse, \(f^{-1}: T_2 \to T_1\).
Let \(U \subseteq T_1\) be any open subset.
\({f^{-1}}^{-1} (U) = f (U)\), which is open on \(T_2\), because \(f\) is open.
So, \(f^{-1}\) is continuous.
So, \(f\) is a homeomorphism.
