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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 Levitt 

   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.