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 $\mathbb{R}^{n}$.

In the version of Lee I own he answers your question with Exercise 1.1 which asks to show that the locally Euclidean condition is equivalent to asking that every point has a neighborhood homeomorphic to an open ball in $\mathbb R^n$ or just $\mathbb R^n$ itself. This condition is clearly stronger than the one originally given. To show the other direction, let $M$ be a topological manifold and let $p \in M$. Suppose $p$ has a neighborhood $U_p$ homeomorphic to $U \in \mathbb R^n$ open by the homeomorphism $\varphi: U_p \to U$. Then there is some open ball $V \subset U$ containing $\varphi(p)$. Then if we restrict $\varphi$ to $\varphi^{-1} (V)$, (which is open since $\varphi$ is a homeomorphism) we have that $\varphi$ maps $\varphi^{-1} (V)$ homeomorphically onto $V$, an open ball in $\mathbb R^n$. 
The next part of the problem just requires showing that open balls in $\mathbb R^n$ are homeomorphic to $\mathbb R^n$, which I'll leave to you. You can explicitly write down the homeomorphism without too much trouble.
