How to prove the inclusion is a homotopy equivalence?

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Homework Help Overview

The discussion revolves around proving that the inclusion of a subspace A into a space X is a homotopy equivalence, given that X deformation retracts to A in a weak sense. The original poster presents a specific scenario involving a homotopy defined on X and seeks guidance on how to proceed with the proof.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to show that the composition of a homotopy with the inclusion map is problematic due to differing domains and codomains. Other participants question the necessity of the inclusion being defined on the image of the homotopy.

Discussion Status

The discussion is ongoing, with participants exploring the implications of the definitions involved and the requirements for the inclusion map. There is a clear expression of uncertainty regarding the composition of the homotopy and the inclusion, indicating that further clarification is needed.

Contextual Notes

Participants are grappling with the definitions of the maps involved and the conditions under which they can be composed. The original poster's concern about the domains and codomains of the functions is a focal point of the discussion.

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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.

Homework Equations





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!
 


qinglong.1397 said:
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?
 


fzero said:
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?
 

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