Poincare Project: More on Compactness

Previously, we defined what it means for a topological space to be compact. The definition ("every open covering has a finite subcovering") is precise but hard to get an intuitive understanding of (well, it was hard for me).

I found it helpful to have some examples of well-known spaces and whether they are compact or not:

• 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.
• a sphere IS compact.

There is an informal sense in which non-compact spaces keep on going, whereas compact spaces stop (or return you to where you started).

Within the context of the Poincaré Conjecture, we will largely be narrowing the spaces we are interested in to those that are compact.

UPDATE: next post