levitt%aids-unix@sri-unix.UUCP (03/19/84)
From: Tod Levitt <levitt@aids-unix> From: ihnp4!houxm!hou2g!stekas @ Ucb-Vax A plane and sphere are NOT topologically equivalent, a sphere has an additional point." More to the "point", the topological invariants of the plane and the (two-) sphere are different, which is the definition of being topologically inequivalent. For instance, the plane is contractible to a point while the sphere is not; the plane is non-compact, while the sphere is compact; the homotopy and homology groups of the plane are trivial, while those of the sphere are not. A more general form of the four-color theorem asks the question: for a given (n-dimensional) shape (and its topological equivalents) what is the fewest number of colors needed to color any map drawn on the shape.