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 PoincarĂ© Conjecture, we will largely be narrowing the spaces we are interested in to those that are compact.

**UPDATE**: next post

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