While I was going through the intro to Hatcher's book on Algebraic Topology there is an exercise which shows that not all contractible spaces allow (strong) deformation retracts onto a point.
I know this may seem like a duplicate question, but please bear with me. Essentially I explicitly am not looking for the answer to the questions already posted.
Part (a) of the question constructs a contractible space $X$, namely the subspace of $\mathbb{R}^2$ formed by point $(x,0)$ with $x\in [0,1]$ and points $(r,y)$ with $0\leq r\leq 1$ rational and $y\in [0,r-1]$. This space deformation retracts onto any point $(x,0)$ in the "base", but not to any other point.
Then comes part (b):
(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.
This question can be done by essentially constructing a weak deformation retract onto the zigzag in the middle of the comb and using a result from a previous exercise.
However then comes part (c):
(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 $Z$
Since this is really a separate question and not say a hint to part (b) it seems like one is supposed to have found some other way to show that $Y$ is contractible that does not explicitly involve constructing a weak deformation retract onto $Z$.
As far as my own ideas I have been able to construct a deformation retract from the left side onto a piece of "base" by essentially concatenating infinitely many copies of the deformation retract of $X$ onto its "base" (I can provide details if necessary). This means that instead of considering the space $Y$ as infinite on both sides we may also look at the space that is only infinite to the right. Beyond that I considered using the results on collapsing contractible subspaces, but $Y$ is not a CW complex and I don't see how I could prove that $(Y,Z)$ satisfies the homotopy extension property.
So my question in the end is: Is there a way to show contractability of $Y$ without explicitely constructing a weak deformation retract onto $Z$, or am I reading too much into the sequencing of the questions?