402: Map of Quotient Topology Is Quotient Map

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A description/proof of that map of quotient topology is quotient map

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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 quotient topology on set with respect to map.
• The reader knows a definition of quotient map.

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Target Context

• The reader will have a description and a proof of the proposition that the map of any quotient topology is a quotient map.

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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 space, $T$, any set, $S$, any surjection, $f:T\to S$, and the quotient topology on $S$ with respect to $f$, $O$, $f$ is a quotient map.

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2: Proof

$f$ is obviously a continuous surjection by the definition of quotient topology. For any subset, $U\subseteq S$, if $f^{-1}(U)$ is open, $U$ is open by the definition of quotient topology.

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References

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