400: Quotient Topology on Set with Respect to Map

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A definition of quotient topology on set with respect to 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: Definition
• 2: Note

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of surjection.

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Target Context

• The reader will have a definition of quotient topology on set with respect to 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: Definition

For any topological space, $T$, any set, $S$, and any surjection, $f:T\to S$, the topology on $S$ defined such that any subset, $U\subseteq S$, is open if and only $f^{-1}(U)$ is open

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2: Note

The quotient topology is indeed a topology, because $f^{-1}(\mathrm{\varnothing })=\mathrm{\varnothing }$, so, $\mathrm{\varnothing }\subseteq S$ is open; $f^{-1}(S)=T$, so, $S\subseteq S$ is open; for any possibly uncountable number of open sets, $\{U_{\alpha }\}$, $f^{-1}({\cup }_{\alpha }{U}_{\alpha })={\cup }_{\alpha }{f}^{-1}(U_{\alpha })$ by the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets, which is open; for any finite number of open sets, $\{U_i\}$, $f^{-1}({\cap }_i{U}_i)={\cap }_i(f^{-1}(U_i))$ by the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets, which is open.

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References

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