Doubt in the definition of manifolds? Let $M$ be a Hausdorff space. Assume that there exists an open covering $\{U_{\alpha} : \alpha \in A\}$ of $M$ and homeomorphisms $\varphi_{\alpha}$, from $U_{\alpha}$
onto an open subset $\varphi_{\alpha}(U_{\alpha})$ of $\mathbb{R}^m(\alpha)$ with $m(\alpha)$ a nonnegative integer.
There exists $k \in \mathbb{N} \cup \infty$ such that whenever for $\alpha,\beta \in A$ we have then the map $U_{\alpha} \cap U_{\beta} \ne \phi$ then the map:
$$\varphi_{\alpha} \circ \varphi_{\beta}^{-1}:\varphi_{\beta}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\alpha}(U_{\alpha} \cap U_{\beta})$$ is $C^k$.
The intuition of the definition behind the manifold is that it behaves locally as a subset of $\mathbb{R}^n$  ( there exists a homeomorphism between $U_p$ a neihbourhood of all possible points $p \in M $ such that $\varphi(U_p)$ is a subset of $\mathbb{R}^n$) such that any function $f:U \to \mathbb{R}$ is $C^k$ [($f \circ \varphi^{-1}):\varphi(U) \to \mathbb{R} $ is $C^k$($k$ times differentiable)].
I can understand the first line of the definition but I am stuck with $$\varphi_{\alpha} \circ \varphi_{\beta}^{-1}:\varphi_{\beta}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) is C^k $$ - how is it related?
 A: Okay. To understand the intuition you need to have the notion of smooth maps from a manifold to another. Notice that you do not(yet) have the notion of differentiability on the manifold but you do have on $\mathbb{R}^{n}$. So to define "smoothness" you have to sort of bring it to $\mathbb{R}^{n}$.
Now a map $F:M\to N$ is smooth when for a point $p$ and charts $(U,\phi)$ of $p$ and $(V,\psi)$ of $F(p)$ , $\psi\circ F\circ \phi^{-1}$ is a smooth map from $\phi(U)\subset \mathbb{R}^{m}\to \mathbb{R}^{n}$  . $m$ and $n$ are dimensions of $M$ and $N$.
So now what can you say about the smoothness of identity map?
The definition is made in such a way such that there is no dependence on these charts when we talk about smoothness . That is , if it is smooth in one chart, it must be so in other charts. You can look at my answer here . In other words, this definition ensures that a bare minimum map like the identity(Which is smooth as a map from euclidean spaces to itself) is a smooth map from a manifold to itself.
