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