Register to reply

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

by aaaa202
Tags: connected
Share this thread:
aaaa202
#1
Feb15-12, 04:15 PM
P: 1,005
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?
Phys.Org News Partner Science news on Phys.org
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief
Fredrik
#2
Feb15-12, 05:00 PM
Emeritus
Sci Advisor
PF Gold
Fredrik's Avatar
P: 9,540
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
#3
Feb15-12, 07:08 PM
Sci Advisor
HW Helper
P: 2,020
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].)


Register to reply

Related Discussions
Windows 7 internet connectivity Computers 7
Ti-89 connectivity Calculators 8
Connectivity Of Graphs General Math 3
Connectivity of Graphs General Math 0