This is my first time to study on the manifold.
I've studied that the topological manifold is:
- locally homeomorphic to Euclidean space, and
- second countable
With these 3 conditions, manifold is one of generalization of Euclidean space since we don't need a full Euclidean, or full flat space, but only local features can also give a bunch of good properties as well as in Euclidean space. Thus, I believe we can deal with differentiation and integration on manifold.
But my question is more fundamental. Why those three conditions are required to define a manifold well?
I think only second condition (locally Euclidean) is needed to satisfy our demands of generalization.
Why are Hausdorff and second countable included?
I admit I can answer this question when if I study futher about manifold. Probably some good properties of manifold will be come out from those two conditions.
But before studying deeply, I want to understand intuitively. I am very unconfortable and cannot accept naturally about this definition.
Someone please give a good explanation. Thanks.