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What does boundary mean in this context then? Because the mobius strip has a boundary on the sides even if it is continuous. I guess boundary means an edge in all directions?

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A mobius strip is an example of a space that is finite but unbounded in a single direction. A two dimensional plane that is finite but unbounded in all directions is called a Kline bottle, but you can’t build them for real in 3D space, only distorted approximations.

You don’t have to do such geometric contortions though. The surface of a sphere is a two dimensional space that is finite but unbounded.

Not sure why you need a Mobius strip when an untwisted rectangle with two opposite edges joined (outside of a cylinder) has the same property. Unlike a sphere that would be a flat 2d space (but unlike a sphere it would be unbounded only in one direction)

Donuts are another option. That’s what you get when you join a square at each pair of opposite edges.

Indeed the gaussian curvature in a torus is 0.

(in 4d only)

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