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

by aaaa202
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aaaa202
#1
Feb15-12, 04:15 PM
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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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Fredrik
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Feb15-12, 05:00 PM
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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\}##.
morphism
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Feb15-12, 07:08 PM
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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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