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The following image is taken from Hatcher's Algebraic Topology on page 2 enter image description here

I want to accept Hatcher's visual argument that the three figures on the right are deformation retracts of a closed ball with two smaller open balls removed, but because I can't formally prove it, even though it intuitively seems obvious I can't accept it.

Below are my attempts to make rigorous and formalize what Hatcher is trying to say:

Hatcher already outlines a sketch of an argument in the previous page on how to construct such a deformation retract

To take an everyday example, the letters of the alphabet can be written either as unions of finitely many straight and curved line segments, or in thickened forms that are compact regions in the plane bounded by one or more simple closed curves.

In each case the thin letter is a subspace of the thick letter, and we can continuously shrink the thick letter to the thin one. A nice way to do this is to decompose a thick letter, call it X, into line segments connecting each point on the outer boundary of X to a unique point of the thin subletter X, as indicated in the figure.

Then we can shrink X to X by sliding each point of X−X into X along the line segment that contains it. Points that are already in X do not move.

From this the general argument I can extract is that any compact region in the plane can be deformation retracted to a subspace that has some specific properties which the subspaces in the right three figures all possess.

Firstly what exactly is this specific property that all the subspaces in these three figures possess?

Secondly how can I prove that a compact subset, which I'll call $X$, of $\mathbb{R}^2$ (endowed with the usual topology) can be decomposed into line segments connecting each point of the boundary of $X$ to a unique point of the subspace $A \subseteq X$ which contains this specific property that I've asked about above?

Finally how can I explicitly write down such a deformation retract $F$ of $X$ onto $A$?


The reason I've had to ask the above three general questions (which rely crucially on the first) is because Hatcher does not give any explicit description of the subspaces (which I'll call $A_1, A_2$ and $A_3$ respectively from left to right) of the three topological spaces on the right which the same space $X$ is being deformation retracted onto.

So I'm guessing there must be some general property that $A_1, A_2$ and $A_3$ all share (all I can think about for now is that they are path connected) which allows us to make this general argument without needing to explicitly construct deformation retracts for these three subspaces.

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(Partial) answer to the frist question: An important necessary condition for those subspaces to be deformation retracts is that the fundamental group must remain the same. More precisely, the inclusion should induce an isomorphism on the fundamental groups.

Anser to the second question: Getting much more general statements will be difficult, I think. In your second question, you seem to assume that $X$ is a very nice space, like that in the drawing of Hatcher. But $X$ could be as complicated as a Cantor set, for example.

Answer to the third question: I understand what you say about not accepting the hand-wavy argument he gives without a rigorous proof. But I think Hatcher is only trying to give some intuition here. You can't have a rigorous proof unless you define the problem properly and rigorously too, and those pictures are just pictures. If your "space" is just a sketch on the paper, your "proof" will at most be another sketch on the paper. I don't know if I am making my point clear. To write down a rigorous proof with formulas you need to define your subset of the plane also with formulas or with some other formal description.

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Hint: the diagrams show you the curves along which the deformation retraction shrinks the space onto its retract (the subspace given by the bold lines) and from this (with some work on your part to parametrise the curves) you can get an explicit definition of a deformation retraction if you feel you need one.

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