Definition of Compatible Charts in Loring Tu

Question

Loring Tu gives the following definition of compatible charts in his book An Introduction to Manifolds (Second Edition, page no. 49).

Definition 5.5. Two charts $(U, \phi:U \to \mathbb{R}^n)$, $(V, \psi:V \to \mathbb{R}^n)$ of a topological manifold are $C^{\infty}$-compatible if the to maps $$\phi \circ \psi^{-1}: \psi(U \cap V) \to \phi(U \cap V), \quad \psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V) $$ are $C^{\infty}$. These two maps are called the transition functions between the charts. If $U\cap V$ is empty, then the two charts automatically $C^{\infty}$-compatible.

My Question

1. Which definition of smoothness is used for these transition functions?

   I am asking this because smooth functions between two manifolds are defined in terms of infinite differentiability in the latter part of the book.

2. Why are the two charts automatically $C^{\infty}$-compatible, if $U\cap V$ is empty?