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


by aaaa202
Tags: connected
aaaa202
aaaa202 is offline
#1
Feb15-12, 04:15 PM
P: 991
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
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
Fredrik
Fredrik is offline
#2
Feb15-12, 05:00 PM
Emeritus
Sci Advisor
PF Gold
Fredrik's Avatar
P: 8,989
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
morphism is offline
#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