Closed Manifolds
I said previously that we were ready to state the Poincaré Conjecture, but there's one more bit of terminology I want to get out of the way and that is closed manifold.
A closed manifold is a compact manifold without a boundary.
We previously listed the following as examples of spaces that are or are not compact:
- the real numbers (under the order topology) is NOT compact.
- any open interval of the real numbers is NOT compact.
- any closed interval of the real numbers IS compact.
- a circle IS compact.
The non-compact examples have the characteristic that you can "keep on going" and keep getting new points whereas the compact examples have the characteristic that you reach a point where there is no more, either because you've reached the edge (i.e. boundary) or because you've gone back to a point you've already been.
Saying without a boundary further restricts us to cases like the circle and not like the closed interval.
So, in other words:
- any open interval of the real numbers is NOT compact.
- any closed interval of the real numbers is COMPACT but has a BOUNDARY.
- a circle is COMPACT and WITHOUT BOUNDARY, i.e. is CLOSED
NOTE: "Closed" here doesn't mean the same thing as a closed subset (i.e. one whose complement is an open set in a topology).
Here are some things to think about:
- All connected one-dimensional closed manifolds are homeomorphic to the circle (or 1-sphere).
- This isn't true for other dimensions because of the possibility of being connected but not simply connected
UPDATE: next post
Comments (4)
David Feldman on Oct. 28, 2005:
Ok I answered my own question by looking at Munkres. A boundary is really only defined on a subset of a topological space with the same topology. Then Munkres defines it as the intersection of the closure of the set with the closure of its complement.
I don't see an obvious way to use this idea to talk about whether the circle has a boundary or not, so I don't yet understand your definition of a closed topological space.
Any ideas?
DF
David Feldman on Oct. 31, 2005:
Ok I get it. Compact.
rms on Sept. 22, 2006:
Manifolds with boundary are not strictly manifolds, because their boundary points *don't* have locally Euclidean neighborhoods. Thus the endpoints of [0,1] have neighborhoods homeomorphic to "half space" H^1 = [0,infinity) rather than full R^1.
So the definition of manifold is extended to "manifold with boundary" that allows points with "half-space" neighborhoods instead of full-space neighborhoods. Such points are (by definition) the boundary points. They correspond to the topological boundary of the manifold when it's embedded in a higher-dimensional space.
"Closed manifolds" by definition have no boundary points; they are true manifolds (w/o boundary) that are compact.
Last Modified: Aug. 9, 2007
Author: James Tauber
David Feldman on Oct. 28, 2005:
Ok ... here is my last post for the night.
I should probably remember this, but I don't. What, exactly, is a boundary. For example, if I put a topology on the interval [0,1] where the open sets are generated by three kinds of sets: [0,a), (a,b) and (b,1] with 0<a<b<1 then I don't have quite the ordinary topology on [0,1]. Would this object have a boundary in your sense? I'm not even sure it would actually be a topology ... it has been a while.
DF