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From: levy@princeton.UUCP
Newsgroups: net.math
Subject: Re: Projection spaces (topology)
Message-ID: <86@princeton.UUCP>
Date: Tue, 27-Sep-83 19:43:02 EDT
Article-I.D.: princeto.86
Posted: Tue Sep 27 19:43:02 1983
Date-Received: Wed, 28-Sep-83 22:41:33 EDT
References: <2683@umcp-cs.UUCP>
Organization: Princeton University
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Identifying each two diametrically opposite points of the boundary of
a closed disc gives the projective plane.  This is easily seen by considering
the disc to be the top half of the two-dimensional sphere; the projective
plane is by definition what you get when you identify opposite points in the
sphere.
You also get the projective plane when you sew together a Mobius strip and
a closed disc along the boundary.
You do indeed get a Klein bottle when you sew together two Mobius strips
(just think of a "symmetric" Klein bottle and split it up into two equal 
parts).
As for identifying the opposite points on the surface of a closed three-
dimensional ball, you get... the three-dimensional projective space!  It's
like the first paragraph above: the three-ball is the "top" half of the
three-sphere, and identifying opposite points on the three-sphere gives you
(by definition) the three-dimensional projective space.
This space is encountered in practice as the space of all possible orientations
you can give to a set of three coordinate axes in Euclidean 3-space; remember
the Euler angles?

Always glad to help,
			-- Silvio Levy