basic doubt about topological manifold In his book "Introduction to Smooth Manifolds", J.M. Lee defines a topological manifold to be a second countable, Hausdorff space with every point having a neighbourhood homeomorphic to an open subset of $\mathbb{R}^{n}$ for some $n$. I was wondering should it not say that it should be homeomorphic to a  connected open subset of $\mathbb{R}^{n}$? He also mentions that equivalent definitions are obtained if one replaces open subset by whole of $\mathbb{R}^{n}$ or by the open unit ball.
I cannot see how this can happen as a connected open subset cannot be homeomorphic to open unit ball or whole of $\mathbb{R}^{n}$.

Well, as I was typing the question I got an insight! We could always choose the connected component of the open set of the manifold which is homeomorphic to an open subset of Euclidean space!
Should this question be closed?
