234: For Injective Closed Map Between Topological Spaces, Inverse of Codomain-Restricted-to-Range Map Is Continuous

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A description/proof of that for injective closed map between topological spaces, inverse of codomain-restricted-to-range map is continuous

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of continuous map.
• The reader admits the proposition that for any bijection, the preimage of any subset under the inverse of the map is the image of the subset under the map.
• The reader admits the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous.
• The reader admits the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset.

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Target Context

• The reader will have a description and a proof of the proposition that for any injective closed map between any topological spaces, the inverse of the codomain-restricted-to-range map is continuous.

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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.

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Main Body

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1: Description

For any topological spaces, $T_1$ and $T_2$, any injective closed map, $f:T_1\to T_2$, and $f$'s restriction on the codomain, $f^{\prime }:T_1\to f(T_1)$, $f^{\prime -1}:f(T_1)\to T_1$ is continuous.

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2: Proof

For any closed set, $C\subseteq T_1$, is ${f^{\prime -1}}^{-1}(C)$ closed on $f(T_1)$? As $f^{\prime }$ is a bijection, ${f^{\prime -1}}^{-1}(C)=f^{\prime }(C)$, by the proposition that for any bijection, the preimage of any subset under the inverse of the map is the image of the subset under the map. As $f$ is closed, $f(C)=f^{\prime }(C)$ is closed on $T_2$, and $f^{\prime }(C)=f^{\prime }(C)\cap f(T_1)$ is closed on $f(T_1)$, by the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset. By the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous, $f^{\prime -1}$ is continuous.

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References

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