269: Inverse of Closed Bijection Is Continuous

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A description/proof of that inverse of closed bijection 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 bijection.
• The reader knows a definition of closed map.
• The reader knows a definition of continuous 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.

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Target Context

• The reader will have a description and a proof of the proposition that the inverse of any closed bijection 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, $T_1,T_2$, and any closed bijection, $f:T_1\to T_2$, the inverse, $f^{-1}:T_2\to T_1$, is continuous.

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2: Proof

For any closed set, $C\subseteq T_1$, $(f^{-1}{)}^{-1}(C)=f(C)$, which is closed by the definition of closed map. By the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous, $f^{-1}$ is continuous.

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References

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