# How to prove the inclusion is a homotopy equivalence?

How to prove the inclusion is a homotopy equivalence???

## Homework Statement

A deformation retraction in the weak sense of a space X to a subspace A is a homotopy $$f_t: X\rightarrow X$$ such that $$f_0=Id_x, f_1(X)\subset A,$$ and $$f_t(A)\subset A$$ for all t. Show that if X deformation retracts to A in this weak sense, then the inclusion $$A\hookrightarrow X$$ is a homotopy equivalence.

## The Attempt at a Solution

I have one problem. Let $$r: X\rightarrow A$$ be the retraction. I can prove that $$ri\simeq Id_A$$. Then I have to prove that $$ir\simeq Id_X$$. Then there is a problem. To achieve this goal, I have to use the homotopy $$f_t$$. This homotopy is defined on the domain X. It seems that I have to compose $$f_t$$ with inclusion, but the codomain of $$f_t$$ is different from the domain of $$i$$. So I do not know how to do. Please give some hints. Thanks a lot!:shy:

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No one knows the answer? What a pity! Hope someone can help me!!!

fzero
Homework Helper
Gold Member

It seems that I have to compose $$f_t$$ with inclusion, but the codomain of $$f_t$$ is different from the domain of $$i$$.
Isn't it more important that $$i$$ be defined on the image of $$f_t$$?

Isn't it more important that $$i$$ be defined on the image of $$f_t$$?
Can $$i$$ be defined on the image of $$f_t$$?

I do not think so. That is because the domain of $$i$$ and the codomain of $$f_t$$ are different sets. For example, if $$t=0$$, $$f_0=Id_X$$ whose codomain is $$X$$, but the domain of $$i$$ is $$A$$. So the composition $$if_0$$ makes none sense.

Is there someone who can help me?