389: If Preimage of Closed Set Under Topological Spaces Map Is Closed, Map Is Continuous

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A description/proof of that if preimage of closed set under topological spaces map is closed, 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 continuous map.
• The reader knows a definition of closed set.

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Target Context

• The reader will have a description and a proof of 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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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 map, $f:T_1\to T_2$, if for any close set, $C\subseteq T_2$, $f^{-1}(C)$ is closed, then $f$ is continuous.

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2: Proof

Suppose that for any close set, $C\subseteq T_2$, $f^{-1}(C)$ is closed. For any open set, $U\subseteq T_2$, $T_2\setminus U$ is closed. $f^{-1}(T_2\setminus U)$ is closed. $f^{-1}(U)=T_1\setminus f^{-1}(T_2\setminus U)$, open. So, as the preimage of any open set is open, $f$ is continuous.

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References

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