Homotopy of Closed Curves on a Simply Connected Region

[IniquiTrance]

Why is it that a region in space is simply connected even when the origin is removed?

Can't one create a closed curve in say the xy plane, centered on the origin, which then cannot be shrunk to a point? (the origin)

Thanks

 

[Jose27]

Because you can move such curve a "little upwards" (or downwards) and then shrink it. This obviously can't be done in the plane and the most immediate analogue would be if you take away, for example, the z-axis.

 

[quasar987]

Why is it that a region in space is simply connected even when the origin is removed?

Can't one create a closed curve in say the xy plane, centered on the origin, which then cannot be shrunk to a point? (the origin)

Thanks

Of course, if the "region" in space is in fact lying in a a plane, then removing a point in the plane makes the region not simply connected, as you say.

But if your simply connected region R has non empty interior and 0 lies in that interior, then R\{0} is still simply connected for the reason indicated by Jose27.

 

[wofsy]

Why is it that a region in space is simply connected even when the origin is removed?

Can't one create a closed curve in say the xy plane, centered on the origin, which then cannot be shrunk to a point? (the origin)

Thanks

Draw a sphere around the point removed from space. Using the radius lines slide your curve onto the sphere. If the curve does not cover the entire sphere then it can be further slid to a point along great circles. iF the curve entirely covers the sphere then you must show that it is homotopic to one that does not. This is a little hard.