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As the title says, why a spiral is the deformation retract of a plane?.

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How do you define a spiral? –  Phira Oct 28 '11 at 17:50
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Can you show that a half line is? Now conjugate that retraction with the homeomorphism that twists the entire plane such that the half line becomes a spiral. –  Henning Makholm Oct 28 '11 at 17:51
    
@HenningMakholm And what if the spiral is doubly infinite? –  Phira Oct 28 '11 at 18:08
    
@Phira, I'm implicitly assuming that it's an ordinary one-armed logarithmic or Archimedean spiral (including the center point because otherwise we're in trouble). If it has multiple arms, start instead by retracting into the union of the appropriate number of half lines radiating out from the origin. –  Henning Makholm Oct 28 '11 at 18:18
    
I'm sorry I do not describe it clearly. Yes, it is an ordinary one-armed Archimedean spiral including the center point. I think it is easy to see that a plane can retract to a spiral without the center.But with a center, I find it not easy.. –  hxhxhx88 Oct 29 '11 at 0:29

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