I could paint the inside of your surface red and the outside blue (where by "inside" I mean the part you've trapped air in). That's not the case with Cliff's bottles.
The abstract object doesn't, the 3d manifestation of it does.
It's just sealing the local minimum. (Maximum? I don't really know what the sign should be for this. Probably maximum, for enclosed volume, since I'm not sure if it's reasonable to talk about the topology having a minimum in this context...)
I don't think I can reason clearly about the 4 dimensional object, part of pointing out that a 3d manifestation does not have the same properties is me trying to do that.