On Saturday, January 25, 2020 at 5:52:05 PM UTC-7, Lawrence Crowell wrote: > > 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 >
Why do cosmologists say a hyper-spherical universe is closed, whereas a plane is open, when in point-set topology they're both closed (both contain their accumulation points)? What do open and closed mean? AG > > 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/430e438f-c3e4-4453-951c-bb58623f7764%40googlegroups.com.
