5 ms·
Even then it doesn’t feel like a great example since it’s not obvious to me that stirring should be a continuous function. E.g. if I put half the glass into a c
by kevinventullo 5y ago
Even then it doesn’t feel like a great example since it’s not obvious to me that stirring should be a continuous function. E.g. if I put half the glass into a container and then pour the container back into the glass, that is definitely a discontinuous operation!
- drdeca 5y agoBut stirring doesn’t break the connnectedness of the water like that?
- kevinventullo 5y agoI guess wherever you dip the spoon in will break it in a kind of discontinuous way?
- drdeca 5y agoIf the spoon was one of those spoons with holes in the, non-handle-part, then I think it would have to produce a discontinuity. But for a normal spoon shape, I don’t think dipping it in makes the map between points in the water fail to be continuous. I’m somewhat confident that there is a smooth path from an embedding of a cylinder to R^3 to an embedding of a cylinder which is a little taller and has a spoon shape removed from it, into R^3 .
- alanbernstein 5y agoI wonder if there is some idealized definition of "stirring" being used here. It seems that actual stirring could conceivably result in any possible rearrangement of particles. As a simple example, which is illustrative even though it seems unlikely as a result of stirring: leave the (x, y) coordinates of each particle the same, while changing the z coordinate as z -> z+k mod h. The result is effectively a discontinuous map (violating the theorem), even though the path of each individual particle must be continuous for physical reasons. Two different notions of continuity here.