The "zig-zag comb" weakly deformation retracts to the central zig-zag line

Question

I am starting to read Hatcher's book on Algebraic Topology, and I am a little stuck with exercise 6 in Chapter 0.

Let $Z$ be the zigzag subspace of $Y$ homeomorphic to $\mathbb{R}$ indicated by the heavier line in the picture:

(see here for picture and definitions)

Show there is a deformation retraction in the weak sense of $Y$ onto $Z$, but no true deformation retraction.

It's easy to show no true deformation retract is possible, but how does one show that a weak deformation retract is possible? Clearly we must deformation retract onto a disconnected subspace of of $Z$; however, it would appear that all open neighborhoods of every point are disconnected.

Answer 1

HINT

In short, imagine that everything 'flows' to the right (and maybe up or down, depending on where it is), down each of the comb bits.