247: For Quotient Map, Induced Map from Quotient Space of Domain by Map to Codomain Is Continuous

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A description/proof of that for quotient map, induced map from quotient space of domain by map to codomain 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 quotient map.

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Target Context

• The reader will have a description and a proof of the proposition that for any quotient map, the induced map from the quotient space of the domain by the map to the codomain 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 quotient map, $f:T_1\to T_2$, the map, $f^{\prime }:T_1/f\to T_2$, is continuous, where $T_1/f$ is the quotient space by $f$, which means that $f^{-1}(p)$ for each $p\in T_2$ are identified.

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2: Proof

$f=f^{\prime }\circ f^{″}$ where $f^{″}:T_1\to T_1/f$. For any open set, $U\subseteq T_2$, is ${f^{\prime }}^{-1}(U)$ open? By the definition of quotient topology, the openness is the openness of ${f^{″}}^{-1}({f^{\prime }}^{-1}(U))$, but ${f^{″}}^{-1}({f^{\prime }}^{-1}(U))=(f^{\prime }\circ f^{″}{)}^{-1}(U)=f^{-1}(U)$, which is open, as $f$ is continuous.

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References

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