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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 ℝ indicated by the heavier line in the picture: (see here for picture and definitions: question about an exercise in hatcher's book (algebraic topology)) 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. –  mixedmath Jul 4 '12 at 17:15
    
@mixedmath Oops. Good job you spotted that. Sorry. –  Matt N. Jul 4 '12 at 18:19

1 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.

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