The notations follow Warner's differential topology book.
Manifold structure on a set $X$ is a choice of both a second countable locally Euclidean (Hausdorff) topology for $X$ and a differentiable structure.
Given $\phi: N \rightarrow M$ between two manifolds and is $C^\infty$, if $\phi$ is one-to-one and $d\phi_p$ is non-singular for each $p\in N$, we call the pair $(N, \phi)$ a submanifold of $M$.
Given $(N_1, \phi_1)$ and $(N_2,\phi_2)$ two submanifolds of $M$, they are equivalent if there exists a diffeomorphism $\alpha$ between $N_1$ and $N_2$ such that $\phi_2\circ\alpha = \phi_1$.
Each equivalent class has a unique representative of the form $(A,i)$ where $A$ is a subset of $M$ with a manifold structure such that the inclusion map $i$ is a $C^\infty$ immersion.
Now given $(N,\phi)$ in an equivalent class, we induce a manifold structure on $A$ by requiring $\phi: N \rightarrow A$ to be a diffeomorphism. This is the only manifold structure on $A$ with the property that $(A,i)$ is equivalent to $(N,\phi)$.
I don't quite see why is this manifold structure unique…
Furthermore, the book says given a subset $A$ of $M$ with a fixed topology. Then there is at most one differentiable structure on $A$ such that $(A,i)$ is a submanifold of $M$.
Could you give me some hints on how to prove the differentiable structure is unique, and I think we need to assume that the fixed topology on $A$ is second countable. The book says it is an application of the follow theorem: Suppose $\phi:N \rightarrow M$ is $C^\infty$, that $(P,\psi)$ is a submanifold of $M$ and that $\phi$ factor through $(P,\psi)$, that is, $\phi(N) \subset \psi(P)$. Since $\psi$ is injective, there is a unique mapping $\phi_0$ of $N$ into $P$ such that $\psi\circ \phi_0 = \phi$. and $\phi_0$ is $C^\infty$ if it is continuous; $\phi_0$ is continuous if $\psi$ is an imbedding.
Best Answer
The claim that the manifold structure on $(A,i)$ is the only one making it equivalent to $(N,\phi)$ is a typical instance of transport of structure: If $A$ is given some smooth manifold structure such that $(A,i)$ is equivalent to $(N,\phi)$, then in particular there is a diffeomorphism $\alpha\colon N\to A$ such that $i\circ\alpha = \phi$, which is to say that $\phi$ itself (with its codomain restricted to $A$) is a diffeomorphism from $N$ to $A$ with the given manifold structure. Thus the topology of $A$ is uniquely determined by declaring its open sets to be those of the form $\phi(U)$ for $U$ open in $N$, and the smooth structure of $A$ is uniquely determined by declaring the coordinate maps to be all maps of the form $\psi\circ\phi$, where $\psi$ is any coordinate map for $N$.
The second claim, "given a subset $A$, there is at most one differentiable structure on $A$ such that $(A,i)$ is a submanifold of $M$," doesn't quite make sense the way you stated it, because a "differentiable structure" only makes sense on a topological space, not on a mere set. I can think of three ways of interpreting it.
Statements 1 and 2 are true (and 1 is a special case of 2). This is Theorem 5.32 in my Introduction to Smooth Manifolds, 2nd ed. (The proof, which is left as a problem for the reader, is basically a straightforward application of Theorem 5.29 on restricting the codomain of a smooth map.)
But statement 3 is false. A standard counterexample is a figure-eight curve in the plane: There are two different manifold structures that turn it into a smooth submanifold, each diffeomorphic to an open interval.