It is a Hausdorff space that is locally homeomorphic to the Euclidean plane, but not paracompact. Manifolds are usually defined to be paracompact because this i

It is a Hausdorff space that is locally homeomorphic to the Euclidean plane, but not paracompact. Manifolds are usually defined to be paracompact because this implies the existence of partitions of unity which have many nice implications: http://math.stackexchange.com/questions/98105/why-are-smooth-manifolds-defined-to-be-paracompact http://math.stackexchange.com/questions/98105/why-are-smooth...