Definition of topological manifold

Question

This might be a stupid question, but I was wondering why we define the topological manifold to be Hausdorff and Second countable?

Thanks :)

Answer 1

The key property of a manifold is that it is locally euclidean. However, another desired property is that it is embeddable in euclidean space. Since Hausdorff and second countable are inherited by subspaces, for a manifold to be embeddable in euclidean space it must satisfy both conditions.

Conversely, it has been proven that any locally euclidean, Hausdorff, second countable space can be embedded in euclidean space.

Answer 2

Otherwise you get some strange stuff like the "long line."