Originally posted by adam warlock
It isn't open because the set of the interior points to the surface of a sphere doesn't coincide with the surface of the sphere. Just look at your won link that defines an open set.

I was talking about the surface of the sphere being the space. My understanding is that a set that includes the whole space is by definition both open and closed. The points interior or exterior to the sphere are not in the space and must be ignored.

Also: how would you embed the surface of a sphere in the surface of the sphere itself?
I am not sure what you are asking here.

"If the surface of the sphere is not the space, but the space is three dimensional space, then clearly every point is a boundary point."
Right on!
But doesn't that contradict your earlier claims about a sphere having no boundaries?
[edit] I see you didn't say that, but said it was boundless. But surely you realize that this 'boundlessness' is only withing the surface of the sphere. In other words, if the surface of the sphere is the space, then it has no boundary points. But this would apply equally well to the surface of a cube, or even a flat square.

Or am I possibly making an error with the definition of a 'space'. Must certain rules apply on a space?