Poincare Project: Topological Properties Revisited

Part of the Poincare Project.

Recall that a topological property is one based only on the open sets of a topology and not any other structure. For this reason a topological property is preserved under a homeomorphism. If one topological space has a topological property and another doesn't have that property then the two spaces can't be homeomorphic.

So far we've talked about the following topological properties:

Compactness is enough to topologically distinguish a circle from an open interval. A circle is compact whereas an open interval is not.

Connectedness is enough to topologically distinguish the real line R from the plane R^2 because if you take away a point from R and from R^2 then R is disconnected but R^2 is still connected.

We don't yet have a topological property that can distinguish a sphere from torus. We shortly will and it will be at the heart of the Poincare Conjecture.

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 1 comment I'm in the process of migrating over.