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

1. Feb 15, 2012

### aaaa202

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?

2. Feb 15, 2012

### Fredrik

Staff Emeritus
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: $\{x\in\mathbb R^3:\|x\|<1\}-\{x\in\mathbb R^3: x_1=x_2=0\}$.

3. Feb 15, 2012

### morphism

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 $\pi_1$. (But it is detected by $\pi_2$.)