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From: ljdickey@watmath.UUCP
Newsgroups: net.math
Subject: projection spaces
Message-ID: <5827@watmath.UUCP>
Date: Sun, 25-Sep-83 14:24:27 EDT
Article-I.D.: watmath.5827
Posted: Sun Sep 25 14:24:27 1983
Date-Received: Mon, 26-Sep-83 11:31:53 EDT
Sender: ljdickey@watmath.UUCP
Organization: U of Waterloo, Ontario
Lines: 48

i recognized two of the things you mentioned in your list of things.
Number (2) is a projective line, and (3) is a projective plane.
These things have been around for a long time, long before topology
came into vogue.  

> From james@umcp-cs.uucp
> Newsgroups: net.math
> Subject: Projection spaces (topology)
> Organization: Univ. of Maryland, Computer Science Dept.
> 
> Today we got started on topological projections spaces, which
> are indentification spaces of friendly things like spheres,
> intervals, and mobius strips.  An identification space is made
> by taking a nice, friendly old thing like, for instance, an
> interval, and mapping it by identifying certain points of it
> to be the same (in the new space).  The simplest example would
> be identifying the two endpoints of a closed interval...then you
> get the equivalent of a circle!  That wasn't too exciting, but
> there are some pretty nice ones, like (and I'll just give a
> loose, cut/sew description):
> 
> 	1) Take a unit disc (pancake) and identify the boundary
> (perimeter), thus drawing all the edge up into a single point.
> 
> 	2) Take a circle, and identify any two diametrically opposite
> points as one.
> 
> 	3) Take a ball, and identify any two diametrically opposite
> points on the surface as one.
> 
> 	4) A mobius strip's edge is a closed curve.  A disc's edge is
> a closed curve.  Sew these two edges together.
> 
> 	5) Sew together the edges of two modius strips.
> 
> Now, what do you think you get when try the above 5?  (1) produces a
> sphere.  (2), (3), and (4) produce things which aren't all drawable,
> definitely not buildable, not easy to visualize, and nameless (as far
> as I know).  I think (5) gives you a Klein bottle, but I haven't proved
> it yet.
> 
> Mathematics sure is fun!
> 
>   --Jim O'Toole
-- 
  Lee Dickey, University of Waterloo.  (ljdickey@watmath.UUCP)
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