I need to clear up my confusion on the definition of a smooth manifold. So we say that $M$ is a smooth manifold (of dimension $n$), if $M$ is Hausdorff and if every $x \in M$ is contained in a neighborhood $U$ that's homeomorphic to an n-ball (the pair $\phi, U$ is called a chart), and if two such charts $\phi_1, U_1$, and $\phi_2, U_2$ overlap, then
$$\phi_2 \circ \phi_1^{-1} : \phi_1(U_1 \cap U_2) \to \phi_2(U_1 \cap U_2)$$
is a smooth map.
But I remember my professor proving that a certain space was a smooth manifold by merely finding an atlas (an open covering of the space by charts) such that the above holds. But according to the definition I wrote, this would be insufficient. Can anyone clear up my confusion?