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From: Bacle on 23 Feb 2010 12:05
Hi:
I am trying to show that the Mobius Band M is a
bundle over S^1.
It seems easy to find trivializations for all points
except for the trouble point (1,0).

This is what I have so far:

Let R be the reals, and I<R be the unit interval
[0,1]<I:

We have these maps:

1) p:I-->S^1 , a qiotient map, i.e., we give S^1 the

quotient topology.

2) q: a quotient map on IxR : identify points

(0,y) with (1,-y) . The "identified space"

--i.e., the quotient of IxR by q -- will be the

top space.

3)The Projection map Pi ,from the top space in 2


down to S^1 is defined by :

Pi([s,t])= p([s]')

( where [s]' is the class of s under p )


Then: *****


A trivialization for U= S^1-{(1,0)} is

U itself; Pi^-1(U)=(1,0)xR , is already a

product space.



How do we find a trivialization containing the

problem point (1,0).?
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