Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!watmath!clyde!floyd!harpo!ihnp4!zehntel!hplabs!sri-unix!levitt@aids-unix From: levitt%aids-unix@sri-unix.UUCP Newsgroups: net.ai Subject: more four color junk Message-ID: <153@sri-arpa.UUCP> Date: Sun, 18-Mar-84 23:45:24 EST Article-I.D.: sri-arpa.153 Posted: Sun Mar 18 23:45:24 1984 Date-Received: Fri, 23-Mar-84 08:56:07 EST Lines: 17 From: Tod LevittFrom: 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.