Number of Connected One-Dimensional Manifolds
The current Wikipedia article on Manifolds says that:
- The open interval (0,1) is a one-dimensional manifold without boundary.
- The closed interval [0,1] is a one-dimensional manifold with boundary.
- Every connected one-dimensional manifold is homeomorphic to one or the other of these.
I'm confused by the third statement as I would have thought that the half-open interval (0,1] and the circle are both connected one-dimensional manifolds but that neither of them are homeomorphic to either the open or closed intervals.
What am I missing?
UPDATE (2005-10-03): Looks like the Wikipedia entry is, in fact, wrong in making the third statement above.
UPDATE: next post
Comments (9)
msh210 (Michael Hamm) on Sept. 29, 2005:
To add to my previous comment: Wikipedia is wrong. Perhaps whoever wrote that was thinking of simply connected manifolds (where 'simply connected' is defined differently from 'connected'); the circle is not simply connected.
msh210 (Michael Hamm) on Sept. 29, 2005:
One last thing: Note that the article Manifold at Wikipedia is currently being rewritten at <a href="http://en.wikipedia.org/wiki/Manifold/rewrite"><tt>http://en.wikipedia.org/wiki/Manifold/rewrite</tt></a>.
James Tauber on Sept. 30, 2005:
Why isn't the circle simply connected?
Allan on Sept. 30, 2005:
> Why isn't the circle simply connected?
You can't contract the whole circle to a point -- it has a "hole". RxR is simply connected, RxR minus the center is not. R^n minus the center _is_ connected for n>2.
See the examples at http://en.wikipedia.org/wiki/Simply_connected#Examples especially the comment that S^n is simply connected for n>=2.
James Tauber on Sept. 30, 2005:
I'd better go back through my recent Poincare Project posts to make sure I haven't assumed the simple connectedness of S^1.
Am I right in thinking that there are two homotopy classes on S^1: the path that goes all the way around the circle is the lone path in one class and all other paths are homotopic?
James Tauber on Oct. 1, 2005:
Sorry, correction to my previous comment:
The path that goes all the way around the circle isn't the lone path because you could go round the circle twice, or n-times right?
However:
(a) the path that goes all the way around the circle is not homotopic to the loops that go less than all the way around before "turning around". Right?
(b) my original post is still correct that "that the half-open interval (0,1] and the circle are both connected one-dimensional manifolds but that neither of them are homeomorphic to either the open or closed intervals." Right?
Janos on Sept. 6, 2006:
(a) Right.
(b) If you include "manifolds with boundary" then right. Otherwise the half-open interval will not be a manifold.
rms on Sept. 22, 2006:
The circle has infinitely many homotopy classes -- the class is determined by the number of times (and the direction) the path goes around the circle.
Last Modified: Oct. 2, 2005
Author: James Tauber
msh210 (Michael Hamm) on Sept. 29, 2005:
The circle is certainly a connected, 1-dimensional manifold. It and the open interval are usually listed as the only connected 1-dimensional manifolds, because manifolds with boundary are excluded. If one includes manifolds with boundary, then both the closed interval and the half-open interval muct be included. Hth.