James Tauber's Blog 2005/09/22

blog > 2005 > 09 >

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 : Created on Sept. 22, 2005 : Last modified Oct. 2, 2005 : (permalink)