# Retraction function -> Onto proof

• scorpius1782
In fact, if r is onto, it cannot be 1-1, as there are more elements in X than in A.In summary, the conversation discussed proving that if a retraction function is onto, then it must be onto. The attempt at a solution mentioned the use of the identity function and the fact that for each input in r(x), there is only one output. However, to prove that r is onto, it is necessary to show that for every element in A, there exists an element in X such that r(x) is equal to that element. It was also noted that r need not be one-to-one, as there are typically more elements in X than in A.
scorpius1782
Wrong forum! sorry please move to math!

## Homework Statement

Prove that if ##r : X \rightarrow A## is a retraction, then r is onto.

Please let me know if I did this correctly!

## Homework Equations

##i_A(x)=x## is the identity function of A

## The Attempt at a Solution

If 'r' is a retraction function then there is a function 's' such that ##A \rightarrow X## and ##r \circ s=i_A##

Then regardless of the input we put into ##r(x)## we will always get x out. Meaning that there is one, and only one, result for each element in 'r'. Then if 'r' is a function in ##\mathbb{R}## it will result in one output for each element in ##\mathbb{R}##.

I'm not sure about your proof. To show r is onto, you need to show that for every element a of A, there exists an element x of X, such that r(x) = a.

Note that r need not be 1-1.

## 1. What is a retraction function?

A retraction function is a mathematical concept used in topology to describe a continuous mapping from a space to a subspace, where the subspace is a subset of the original space. It essentially "retracts" the original space onto the subspace.

## 2. How does a retraction function work?

A retraction function works by taking all the points in the original space that lie within the subspace and mapping them onto the corresponding points in the subspace. The remaining points in the original space are left unchanged.

## 3. What is the purpose of a retraction function?

The purpose of a retraction function is to simplify a space by reducing its dimensionality. It can also be used to transform a more complex space into a simpler one, making it easier to study and analyze.

## 4. What is the difference between a retraction function and a deformation retraction?

A deformation retraction is a special case of a retraction function where the subspace is also a deformation of the original space. This means that the subspace can be continuously deformed into the original space without changing the shape of the subspace. A general retraction function does not have this restriction.

## 5. How is a retraction function related to proof in mathematics?

In mathematics, a retraction function is used as a tool for proving theorems and propositions. It can be used to show that two spaces are homeomorphic (have the same shape) or to prove the existence of certain topological properties in a space. It is also used in algebraic topology to study the properties of different spaces and their transformations.

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