Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10 5/3/83; site umcp-cs.UUCP
Path: utzoo!linus!philabs!seismo!rlgvax!cvl!umcp-cs!james
From: james@umcp-cs.UUCP
Newsgroups: net.math
Subject: Projection spaces (topology)
Message-ID: <2683@umcp-cs.UUCP>
Date: Thu, 22-Sep-83 13:36:23 EDT
Article-I.D.: umcp-cs.2683
Posted: Thu Sep 22 13:36:23 1983
Date-Received: Fri, 23-Sep-83 23:29:33 EDT
Organization: Univ. of Maryland, Computer Science Dept.
Lines: 34

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