Is S^2 - {N} Homeomorphic to R^2?

[PsychonautQQ]

Is the sphere minus the north pole homeomorphic to the plane? Obviously no, right? Because the sphere minus a point is compact and then plane is not.

I was just thinking that the one point compactification of the plane is homeomorphic to a sphere, because now you have a way to map to the north pole. So I was just thinking, maybe if we subtract the north pole it will be homemorphic to the plane now. What is wrong with my thinking here?

 

[fresh_42]

Why is the sphere minus a pole closed? And why shouldn't the stereographic projection work?

 

[PsychonautQQ]

Ahh, so the sphere minus a point is not compact, duh. Thanks.

 

[WWGD]

Sort of weird how removing a single point changes so much topologically. Uncountanly-many points in sphere, yet removing a single one changes compactness, closedness ( tho, obviously, not boundedness).