I'm trying to solve part (b) of exercise 0.6 in Hatcher's Algebraic Topology:
(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 has been discussed several times before on this site (and elsewhere). However, the focus was always on proving that the space isn't deformation retractable. What I'm interested in is showing that the space is contractible (identity function is nullhomotopic). There was one question addressing this issue for a similar space, but the answer is beyond my knowledge. I'm convinced there is an easier proof for someone who's only read the first few pages of Hatcher's book.
I found some solutions on the web, but some are wrong and some are missing crucial details.
What I know:
- It's sufficient to show that the space deformation retracts in the weak sense to the bold zigzag. By a previous exercise (0.4), there is a homotopy equivalence. The bold zigzag is homeomorphic to $\mathbb{R}$, which makes it contractible.
- We can construct this space as a quotient space of the disjoint union of countably many copies of one triangle, by identifying common edges. It's easy to show that this union is contractible. I'm having a hard time showing that the homotopy transfers to the quotient space.
This is self-study. Any help would be appreciated. Thanks.