description/proof of that continuous bijection from compact topological space onto Hausdorff topological space is homeomorphism
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of compact topological space.
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of bijection.
- The reader knows a definition of homeomorphism.
- The reader admits the proposition that any closed continuous bijection is a homeomorphism.
- The reader admits the proposition that any closed subset of any compact topological space is compact.
- The reader admits the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is a compact subset of the codomain.
- The reader admits the proposition that any compact subset of any Hausdorff topological space is closed.
Target Context
- The reader will have a description and a proof of the proposition that any continuous bijection from any compact topological space onto any Hausdorff topological space 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:
\(T_1\): \(\in \{\text{ the compact topological spaces }\}\)
\(T_2\): \(\in \{\text{ the Hausdorff topological spaces }\}\)
\(f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\} \cap \{\text{ the bijections }\}\)
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Statements:
\(f \in \{\text{ the homeomorphisms }\}\)
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2: Proof
Whole Strategy: Step 1: apply the proposition that any closed continuous bijection is a homeomorphism.
Step 1:
Let \(C \subseteq T_1\) be any closed subsets.
\(C\) is a compact subset, by the proposition that any closed subset of any compact topological space is compact.
\(f (C)\) is a compact subset, by the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is a compact subset of the codomain.
\(f (C)\) is a closed subset, by the proposition that any compact subset of any Hausdorff topological space is closed.
So, \(f\) is a closed map.
So, by the proposition that any closed continuous bijection is a homeomorphism, \(f\) is a homeomorphism.
