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I must understand connectivity wrong, because my book says this. The

  1. Feb 15, 2012 #1
    I must understand connectivity wrong, because my book says this. The region between to concentric spheres is simply connected? How is this possible when there is clearly a hole in the middle of this region?
     
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  3. Feb 15, 2012 #2

    Fredrik

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    Re: connectivity

    That kind of "hole" doesn't prevent you from continuously shrinking a closed curve to a point. It would have to be a hole shaped like a cylinder or something, that goes all the way through the sphere. Consider e.g. the open unit ball with a line removed: ##\{x\in\mathbb R^3:\|x\|<1\}-\{x\in\mathbb R^3: x_1=x_2=0\}##.
     
  4. Feb 15, 2012 #3

    morphism

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    Re: connectivity

    Is a sphere, which has a hole in the middle, not simply connected? Note that this is really the same as your example, since your region deformation retracts onto a sphere.

    The kind of hole you're noticing doesn't affect simply connectedness - it isn't detected by [itex]\pi_1[/itex]. (But it is detected by [itex]\pi_2[/itex].)
     
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