Unique manifold structure

Question

I am reading the first chapter from the book - Foundations of Differentiable manifolds and Lie groups by Warner. There, he has given two statements to be proved as exercises.

a) Let $M$ be a differentiable manifold and $A$ be a subset of $M$. Fix a topology on $A$. Then there is at most one differentiable structure on $A$ such that $(A,i)$ is a submanifold of $M$, where $i$ is the inclusion map.

b) Again let $A$ be a subset of $M$. If in the relative topology, $A$ has a differentiable structure such that $(A,i)$ is a submanifold of $M$, then $A$ has a unique manifold structure (that is, unique second countable locally Euclidean topology together with a unique differentiable structure) such that $(A,i)$ is a submanifold of $M$.

I was able to prove part a). But I don't understand what to prove in part b). In the relative topology $(A,i)$ becomes a submanifold, so by part a) the differentiable structure is unique. So should I prove, no matter what topology I put on $A$, I will get the same differentiable structure? But the underlying topological spaces will be different. Is that okay? Or should I prove no matter what topology I put, such that $(A,i)$ becomes a submanifold, the two submanifolds are diffeomorphic? And how do I begin proving this? Any help will be appreciated!

Answer 1

Part (a) says that given a topology on $A$, there's at most one differentiable structure on $A$ that makes $(A, i)$ a submanifold of $M$.

It's possible, once you have proved this, that for several different topologies on $A$, there are corresponding differentiable structures that make $(A, i)$ a submanifold.

Part (b) specifies a topology (the relative topology), and says that for this one topology, there is at least one differentiable structure compatible with it, from which you can argue that there's at most one such structure that makes $(A, i)$ a submanifold, reasoning from part "a".

But the question is "might there be some other topology for which $A$ has an differentiable structure, and for which $(i, A)$ is a submanifold?" Or to put it differently: if $(i, A)$ is a submanifold, why must the topology on $A$ be the relative topology rather than some other one?