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Topic: Classification of 1-manifolds (was: 1-manifold homeomeorphic to R)
Replies: 7   Last Post: May 3, 2009 5:50 AM

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Rodolfo Medina

Posts: 15
Registered: 8/21/06
Classification of 1-manifolds (was: 1-manifold homeomeorphic to R)
Posted: Apr 19, 2009 9:31 AM
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Rodolfo Medina <rodolfo.medina@gmail.com> writes:

> Here:
>
> [...]
>
> http://en.wikipedia.org/wiki/Topological_manifold
>
> , `Classification of manifolds', it is stated that a connected 1-topological
> manifold is homeomorphic to R.



Alfredo Restrepo Palacios <arestrep@uniandes.edu.co> writes:

> Hi,up to homeomorphisms, there are only two connected (separable, metric)
> 1-manifolds: the (noncompact) open interval I' and the (compact) circle
> S1. They are both orientable, the "real projective line" is also S1. You can
> also take a look at Christenson and Voxman, Aspects of Topology,



Thanks. I haven't looked at the book you suggest yet. But here:

http://en.wikipedia.org/wiki/Long_line_(topology)

it is stared a more general result: that a 1-manifolds with boundary is
homeomorphic either to R or to the circle, or even to a closed or half-open R
interval.

Can anybody suggest where to find a complete demonstration of this more general
statement? Or maybe the demonstration reported by Prof. Lee can be easily
generalized?

Thanks for any help
Rodolfo




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