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One perspective that might be useful is to think about the functional definition of a straw. The job of a straw is to enclose part of the two-dimensional surfac

One perspective that might be useful is to think about the functional definition of a straw. The job of a straw is to enclose part of the two-dimensional surface of a liquid, then connect the enclosed region to an enclosed container with a lower air pressure than the surroundings. It kind of absolutely follows from those requirements, that a straw has to be some sort of shape with a boundary and a single hole. If it has no boundary, you can't connect it to the low-pressure reservoir; if it has no hole, it can't enclose liquid surface; if it has more holes, it doesn't separate space above the liquid into areas which can have different pressures (hence why a pinhole in a straw stops it from working). There are other forms that could function as a straw - you could have a straw that splits in two and then joins back up (topologically equivalent to a torus pierced by two circles - where a normal straw is a sphere pierced by two circles). Essentially you can take any surface which has an inside and an outside, pierce it twice, and use it as a straw (so you can't do it, for example, with a klein bottle - but you could do it with a trefoil knot). Put your mouth round one piercing, stick the other against the surface of the liquid, and your topologically advanced straw should work. This also kind of points to why it's confusing to talk about how many holes there are in a straw - because if you start with a closed 2D surface (like a sphere), you have to pierce it twice to make a straw. But in topology, the first 'hole' you add isn't a 'hole' at all (you can't loop a string through it, as the video says) - it just opens the closed surface.

ColinWright 8y ago
> ... if you start with a closed 2D surface (like a sphere), you have to pierce it twice to make a straw. But in topology, the first 'hole' you add isn't a 'hole' at all ... That's from one point of view, as the video says. From another point of view it is a hole. From that point of view, making a "straw" from a sphere requires two holes. The question the video then goes on to ask is whether that point of view is "the best" one. The conclusion is that it probably isn't the most useful - in this context. But in other contexts, if I have a sphere - genuinely a two dimensional object, all the points (x,y,z) in R^3 satisfying x^2+y^2+z^2=1, when I then punch a hole in it, it has a hole. It's now homeomorphically a disk, but it's a sphere with a hole in it. In this context, for this purpose. So I think you are overly-condensing the video and removing important points.
amelius 8y ago
Perhaps a hole should be defined by other objects. For example, think of a closed rope (loop) that passes through the straw. You can't remove the loop and the straw from eachother. But, if you imagine the rope to be so thin that it can pass between the atoms of the straw, then you could do it (mathematically speaking, not physically). So the straw has one hole w.r.t. the normal rope, but zero holes w.r.t. the very thin rope.

ColinWright 8y ago

> ... if you start with a closed 2D surface (like a sphere), you have to pierce it twice to make a straw. But in topology, the first 'hole' you add isn't a 'hole' at all ... That's from one point of view, as the video says. From another point of view it is a hole. From that point of view, making a "straw" from a sphere requires two holes. The question the video then goes on to ask is whether that point of view is "the best" one. The conclusion is that it probably isn't the most useful - in this context. But in other contexts, if I have a sphere - genuinely a two dimensional object, all the points (x,y,z) in R^3 satisfying x^2+y^2+z^2=1, when I then punch a hole in it, it has a hole. It's now homeomorphically a disk, but it's a sphere with a hole in it. In this context, for this purpose. So I think you are overly-condensing the video and removing important points.

amelius 8y ago

Perhaps a hole should be defined by other objects. For example, think of a closed rope (loop) that passes through the straw. You can't remove the loop and the straw from eachother. But, if you imagine the rope to be so thin that it can pass between the atoms of the straw, then you could do it (mathematically speaking, not physically). So the straw has one hole w.r.t. the normal rope, but zero holes w.r.t. the very thin rope.