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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: enter image description here

(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.

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    This is not a duplicate of that question. Although the source is the same, neither of the other two formulations actually show a weak reatract, but instead show that no def retract is possible.2012-07-04
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    @mixedmath Oops. Good job you spotted that. Sorry.2012-07-04

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