You have to go beyond point-set topology and consider homology or homotopy theory, With a sphere a curve is a loop that is contractible to a point. A line in flat spacetime is not contractible. This might make one thinkthere is a homology, or cohomology, of H^1(R^2) = ker(M)/im(M) for M a map. The homotopy π_1 for the sphere is zero, contractible, but is Z for the Euclidean space. One might think the homology H_1(R^2) is the same, but Euclidean plane and 2-sphere have a trick up their sleeve with the stereographic projection so the pole of S^2 gets mapped to "infinity" So the middle homology group or ring is zero. The homotopy fundamental form π_1 has commutators that make it not zero. However, that stereographic projections means the point at the pole is mapped away so while the sphere has H_0(S) = H_2(S) = Z the Euclidean plane has H_0(R^2) = 0.
LC On Friday, January 24, 2020 at 7:38:43 PM UTC-6, Alan Grayson wrote: > > Both are connected. Both have no boundary. Both are closed, since both > contain their accumulation points. Both have uncountable elements. So how > can they be distinguished within the context of point-set topology? TIA, AG > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/7cca37d9-4316-4180-a191-28a81764ff96%40googlegroups.com.
