In this post I wrote down the definition of an abstract manifold. Let me repeat it here:
Definition. Let $M$ be metric space and $\left(V_{i}\right)_{i\in I}$ be an open cover of $M$ with open sets $U_i\subset\mathbb R^m$ and homeomorphisms $F_i: U_i\rightarrow V_i$. Then we say $M$ is an (abstract) $m$-dimensional $C^k$ manifold if for all two open sets $V_1$, $V_2\subset M$ with maps $F_1$ and $F_2$ the transition map $$F_{2}^{-1}\circ F_1: \quad F_1^{-1}(V_1 \cap V_2) \rightarrow F_{2}^{-1}(V_1\cap V_2)$$ is a $C^k$ diffeomorphism.
I realized that I have not fully captured yet the part with the transition maps. Why exactly is it $F_{2}^{-1}\circ F_1: F_1^{-1}(V_1 \cap V_2) \rightarrow F_{2}^{-1}(V_1\cap V_2)$ and not $F_{2}^{-1}\circ F_1: U_1 \rightarrow U_2$? After all, the domain of $F_1$ is $U_1$ and the codomain of $F_2^{-1}$ is $U_2$.
Best Answer
The map $F_1$ is a map from $U_1$ to $V_1$. And $F_2^{\,-1}$ is a map from $V_2$ to $U_2$. But $F_2^{\,-1}\circ F_1$ is not defined on the whole $U_1$; $(F_2^{\,-1}\circ F_1)(x)$ makes sense only for those $x\in U_1$ such that $F_1(x)$ belongs to the domain of $F_2^{\,-2}$. And this happens exactly when $x\in F_1^{\,-1}(V_1\cap V_2)$, that is, when $x$ is such that $F_1(x)\in V_1\cap V_2$.