A proper container has two surfaces, one keeping the outside out and one keeping the inside in.
Just like the Moebius Strip, the Klein Bottle only has one surface. If that's the outer surface, then yes, you can pour water into the outside (#1). Or if it's the inner surface, than you can pour water inside (#2). Haha, no, you can't as the water was already on the outside (#1) or on the inside (#2).
You can arbitrarily defines any part of the surface as "inside" or "outside", but this doesn't make sense topologically. It's like taking a chessboard and saying "I call the black squares are 'inside', and the white ones 'outside'."
And yes, you can pour water on a chessboard. Smash the back squares with a mallet first, and they might actually hold water.
What we see when we see a "real-life Moebius Strip" or a "real-life Klein Bottle" are actually imperfect illustrations (and, in case of the Klein Bottle, additionally a 3D projection of a 4D object). Topologically, the strip should have zero thickness, and the bottle walls the same.
Fun experiment: what's the "inside volume" of a bowl? (You can measure it by weighing the mass of water it holds at 4°C, one liter will be 1kg). Now turn it upside down. Has it still the same "inside volume"?
Correct answer for the bowl: it's not how much water it can hold, it's how much water it displaces when it gets fully submerged. That's for a real-life bowl. A theoretical topological bowl doesn't have any volume.
A proper container has two surfaces, one keeping the outside out and one keeping the inside in.
Just like the Moebius Strip, the Klein Bottle only has one surface. If that's the outer surface, then yes, you can pour water into the outside (#1). Or if it's the inner surface, than you can pour water inside (#2). Haha, no, you can't as the water was already on the outside (#1) or on the inside (#2).
You can arbitrarily defines any part of the surface as "inside" or "outside", but this doesn't make sense topologically. It's like taking a chessboard and saying "I call the black squares are 'inside', and the white ones 'outside'."
And yes, you can pour water on a chessboard. Smash the back squares with a mallet first, and they might actually hold water.
What we see when we see a "real-life Moebius Strip" or a "real-life Klein Bottle" are actually imperfect illustrations (and, in case of the Klein Bottle, additionally a 3D projection of a 4D object). Topologically, the strip should have zero thickness, and the bottle walls the same.
Fun experiment: what's the "inside volume" of a bowl? (You can measure it by weighing the mass of water it holds at 4°C, one liter will be 1kg). Now turn it upside down. Has it still the same "inside volume"?
Correct answer for the bowl: it's not how much water it can hold, it's how much water it displaces when it gets fully submerged. That's for a real-life bowl. A theoretical topological bowl doesn't have any volume.