I am tackling Hatcher's Algebraic Topology Problem 6b. But I am wonder if my proof is okay, or that is too descriptive and I need to carry out explicit computation in coordinates? Also, I am not sure about my two-step homotopy construction. I wonder if anyone could spare sometime to help me take a look at it?

(b) Let $Y$ be the subspace of $\mathbb{R}^2$ that is the union of an infinite number of copies of $X$ arranged as in the figure below. Show that $Y$ is contractible but does not deformation retract onto any point.

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Using the construction we did in (a), we can retract $Y$ on to the thick zig-zag line in the middle. We consider each $X$ individually, and $Y$ as a union of an infinite number of $X$ follows.

Assume for contradiction that $Y$ deformation retracts onto some point $y \in Y$, then by the preceding problem, Inclusion Map Is Nullhomotopic, we know that the inclusion map on a neighborhood in $Y$ is homotopic to a constant map. This is not true:

In (a), we proved that it is impossible to retract to the thin lines on $X$. The same argument stand for the thin lies on $Y$.

Now we look into the thick lines on $Y$. Apparently, by the construction given, for each point on the thick line, clearly no matter how small a neighborhood we choose, the neighborhood is always non-path-connected, which clearly can't be homotopy to a constant map.

(c) Let $Z$ be the zigzag subspace of $Y$ homeomorphic to $\mathbb{R}$ indicated by the heavier line in the picture. Show there is a deformation retraction in the weak sense of $Y$ onto $Z$, but no true deformation retraction.

Definition We say that $f: X \to X$ is a deformation retraction of a space $X$ onto a subspace $A \subset X$ if there exists a family of maps $f_t : X \to X$ with $t \in [0,1]$ such that $f_0 = \mathbb{1}$ (the identity) and $f_1 (X) = A$ and also $f_t$ restricts to the identity on $A$ for each $t$.

Definition A weak deformation retraction is almost the same, only that we now relax the conditions $f_1(X) = A$ to $f_1 \subset A$ and, for each $t \in [0,1]$ we require that $f_t(A) \subset A$.

As we showed above in (b), we see that $Y$ is contractible. Hence there is a weak deformation of $Y$. By the same construction we made in (a) retract $X$, the weak deformation is onto $Z$.

There is no true deformation retraction, because if it does, then it must be $Z$ or a subset of $Z$. Either case, would be a deformation retract to a single point. This follows the lemma that the deformation retraction is nullhomotopic. Which is impossible, since any neighborhood of a point on $Z$ is non-path-connected.

Thank you very much!


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