James Tauber's Blog 2005/09/22
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
by James Tauber : Created on Sept. 22, 2005 : Last modified Oct. 2, 2005 : Categories poincare_project : 9 comments (permalink)