In the definition of Manifold, we are given countable atlases $(U_i, \psi_i)$, such that every point $p$ is contained in one such $U_i$ and $\psi_i(U_i)$ is homeomorphic to some subset in $\mathbb{R}^n$.
Now I want to reverse this process. Suppose I choose a countable subset in $\mathbb{R}^n$ and also assume I specify the individual intersection between any two subsets. Now I move around those subsets to satisfy the intersection rule I specified above, then will the resultant figure be a manifold. Also, can every manifold be deconstructed and then reconstructed in the way I mentioned.
Note:
(1) I am allowed to work in higher $\mathbb{R}^n$, as I figured the intersection that I specified might be unattainable in lower dimension but possible in a higher dimension (I will further clarify on this remark if needed. Let me know in the comment).
(2) The subsets are of the same dimension. For example, a circle has dimension 1 because it is essentially a length, the disk has dimension 2 because it is an area and so on.