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Let $X$ be a set.

Let $\mathscr{A}$ and $\mathscr{B}$ be $C^k$-atlases on $X$. Let $\tau_1,\tau_2$ be the canonical topologies on $X$ determined by $\mathscr{A}$ and $\mathscr{B}$ respectively.

I'm curious on the relation between [equivalence of $C^k$-atlases] $\leftrightarrow$ [canonical topologies determined by $C^k$-atlases]

It is true that if $\mathscr{A}$ and $\mathscr{B}$ are equivalent, then $\tau_1=\tau_2$.

What about the converse?

Since topology does not say anything about differentiability, the converse would be false. But what is a countetexample for this?

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    Even though the topology says nothing about smoothness, these topologies would still be induced by each atlas? In this case, I think the converse MIGHT hold. My reasoning goes like this: you can view the manifold as an embedded submanifold of itself since the identity would be a homeomorphism. This makes the identity smoothly universal, implying that any smooth atlas remains a smooth atlas when taken with respect your other atlas. But of course, then they are equivalent. – 2017-02-25
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    Look into [Milnor's exotic spheres](https://en.wikipedia.org/wiki/Exotic_sphere) for one source of counter-examples where $k = \infty$ i.e., the differentiable structures are smooth. – 2017-02-25

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How about this? Take $X = \mathbb R$. My first atlas has a single chart $\varphi_1 : X \to \mathbb R$ with $\varphi_1(t) = t$. My second atlas has a single chart $\varphi_2: X \to \mathbb R$ with $\varphi_2(t) = t^3$. The topologies induced by both atlases are the same, since $\varphi_1 \circ \varphi_2^{-1}$ is a homeomorphism, which means that $\varphi_1(V)$ is open iff $\varphi_2(V)$ is open for any $V \subset X$. But the two atlases are not compatible, since $\varphi_1 \circ \varphi_2^{-1}$ is not smooth at the origin.

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    By the way, in my example, $X$ with the first atlas is diffeomorphic to $X$ with the second atlas; the diffeomorphism is $t \mapsto t^{1/3}$. But the two differentiable structure are not equivalent since the identity map $t \mapsto t$ is not a diffeomorphism. I think Milnor's exotic spheres are examples where two differentiable structures give spaces that are homeomorphic but not diffeomorphic under any diffeomorphism. – 2017-02-25