379: Continuous Surjection Between Topological Spaces Is Quotient Map if Any Codomain Subset Is Closed if Its Preimage Is Closed

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A description/proof of that continuous surjection between topological spaces is quotient map if any codomain subset is closed if its preimage is closed

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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
• 3: Note

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Starting Context

• The reader knows a definition of continuous map.
• The reader knows a definition of surjection.
• The reader knows a definition of closed set.
• The reader knows a definition of quotient map.
• The reader admits the proposition that the preimage of the codomain minus any codomain subset under any map is the domain minus the preimage of the subset.

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Target Context

• The reader will have a description and a proof of the proposition that any continuous surjection between any topological spaces is a quotient map if any codomain subset is closed if its preimage is closed.

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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 continuous surjection, $f:T_1\to T_2$, $f$ is a quotient map if any subset, $S\subseteq T_2$, is closed if its preimage, $f^{-1}(S)$, is closed.

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2: Proof

Let us suppose that $S$ is closed if $f^{-1}(S)$ is closed. For any subset, $S^{\prime }\subseteq T_2$, if $f^{-1}(S^{\prime })$ is open, is $S^{\prime }$ open? $T_1\setminus f^{-1}(S^{\prime })$ is closed. $T_1\setminus f^{-1}(S^{\prime })=f^{-1}(T_2\setminus S^{\prime })$, by the proposition that the preimage of the codomain minus any codomain subset under any map is the domain minus the preimage of the subset. $T_2\setminus S^{\prime }$ is closed. $S^{\prime }$ is open.

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3: Note

Being continuous is required as a precondition, because the continuousness is not guaranteed by just any codomain subset's being closed if the preimage is close.

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References

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