A hollow sphere has two surfaces, an inside and an outside, but can a hollow solid have only one surface and no edges?

Gnomon

The only hollow solid which has one surface and no edges is the Klein surface, often called the Klein bottle. This only achieves this by bending through the fourth dimension, so it doesn't really exist in our three-dimensional world. Jan 20 04, 2:25 AM

gmackematix

A yay for the Klein bottle, Gnomon. If a rectangular strip is twisted and the short ends are joined seamlessly then the resulting shape is a Mobius strip with one surface and one edge. If that edge is joined seamlessly to itself then the resultant hollow 3-D shape must have one surface and no edges. I think that this shape is topologically equivalent to (that is, can be stretched into) a Klein bottle. While the procedures above may be difficult for the glassblower I don't see why a trip into a fourth spatial dimension is necessary? Nor am I sure that such a surface is unique although, topologically, it may be. Jan 20 04, 7:39 PM