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 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:
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:
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.
The original post had 4 comments I'm in the process of migrating over.