Prove that a retraction is a quotient map

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SUMMARY

The discussion focuses on proving that a retraction is a quotient map, specifically addressing the conditions under which the union of two sets is open in a subspace topology. The participant identifies that the preimage of an open set under a retraction, intersected with the subspace, is open in the subspace. However, they express uncertainty regarding the openness of the other unioned set, indicating a need for further clarification on this aspect.

PREREQUISITES
  • Understanding of topology concepts, particularly subspace topology.
  • Familiarity with retraction and quotient map definitions.
  • Knowledge of open sets and their properties in topological spaces.
  • Experience with set operations and intersections in topology.
NEXT STEPS
  • Study the properties of retractions in topology.
  • Research the definition and examples of quotient maps.
  • Learn about open sets in subspace topologies.
  • Examine the intersection of open sets and their implications in topology.
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Mathematics students, particularly those studying topology, and educators seeking to clarify concepts related to retractions and quotient maps.

Jamin2112
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Homework Statement



As in title.

Homework Equations



Described in my attempt.

The Attempt at a Solution



screen-capture-29.png





Where do I go from here? I need to show that those 2 unioned sets are open in A. I'm not seeing it
 
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Wait ... hang on ... I think I might have it.

I know that (r^(-1) (U) ⋂ A) is open in A if we mean with respect to the subspace topology, since it is the intersection of an open set r^(-1) (U) of X with A. Not sure about the other unioned set though. Thoughts?
 
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