Prove that a retraction is a quotient map

In summary, a retraction in topology is a continuous mapping from a topological space to a subspace of itself that fixes all points in the subspace. To prove that a retraction is a quotient map, it must be shown to be surjective and its inverse image of any open set in the subspace must be an open set in the larger space. This is important for understanding the topological structure of the space and constructing new spaces. A retraction must be surjective and continuous to be a quotient map, as it must preserve the topological structure of the space.
  • #1
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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  • #2
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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1. What is a retraction in topology?

A retraction is a continuous mapping from a topological space to a subspace of itself that fixes all points in the subspace, meaning that the image of each point in the subspace is equal to the point itself.

2. How do you prove that a retraction is a quotient map?

A retraction can be proven to be a quotient map by showing that it is surjective, meaning that every point in the subspace has a preimage in the larger space, and that the inverse image of any open set in the subspace is an open set in the larger space.

3. Why is it important for a retraction to be a quotient map?

A retraction being a quotient map allows for a better understanding of the topological structure of the space, as well as providing a useful tool for constructing new spaces from existing ones. It also allows for the study of properties and relationships between spaces by examining the properties of the quotient space.

4. Can a retraction be a quotient map if it is not surjective?

No, a retraction must be surjective in order to be a quotient map. This is because for a quotient map, every point in the quotient space must have a preimage in the original space. If the retraction is not surjective, there will be points in the quotient space without a preimage, violating the definition of a quotient map.

5. Are there any other conditions that a retraction must satisfy in order to be a quotient map?

Yes, in addition to being surjective, a retraction must also be continuous. This means that the inverse image of any open set in the subspace must be an open set in the larger space. If the retraction is not continuous, it may not preserve the topological structure of the space and therefore cannot be a quotient map.

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