If calculus can be applied to non-numbers. Trying to understand differential structure.
So (from Wikipedia) two spaces with a homeomorphism are considered the same. A manifold is a topological space that locally resembles the Euclidean space. A Euclidean space is a space over the real numbers. That is the first question (1) below. A differentiable manifold is a manifold locally similar enough to a linear space to allow one to do calculus. That leads to my next question (2) below. Finally, a differentiable structure on a set makes it into a differentiable manifold, which is a topological manifold with some additional structure to allow doing differential calculus on the manifold.


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*Can a manifold be a topological space outside of the numbers / matrices / complex numbers that has some sort of mapping/relation ("homeomorphism") to the Euclidean space, and in that way be considered "locally resembling the Euclidean space". And in this way calculus can be applied to a topology (non-numbers).

*Or is calculus only done over numbers / matrices / complex numbers, or can calculus be done on arbitrary spaces such as groups, rings, fields, etc. (all generated by non-numbers).


Basically I'm wondering if you can construct a manifold by constructing a topological space without numbers, and what it would look like. If not, then if you can construct a topological space $X$ without numbers, then construct a space $Y$ with numbers, construct a mapping $f : X \to Y$, and then say the topological space is a manifold. Wondering if an example could be outlined showing what this would look like. Wondering if calculus can then essentially be applied on non-numbers in this fashion.
 A: As @Yucoub Kureh said, formal derivatives can be defined for the polynomial ring $R[X]$ or over the ring of formal power series $R[[X]]$. But obviously $R$ itself may have no topological structure and so I am not really so sure if you can call this calculus. For example, one defines formal derivatives over polynomials in $\mathbb{Q}[X]$ in Galois theory but the reason we do this is to show that given any irreducible polynomial, it is a separable polynomial as $f$ and its formal derivative $f'$ are coprime polynomials. The notion of "continuity" or "topology" is not really relevant one defines the formal derivative in this context. 
I guess the inner product over the space of continuous functions is an example where we have a topological structure.  (For example, $C[0,1]=\{f:[0,1]\to\mathbb{R}\text{ such that }f\text{ is continuous}\}$ and we have $<f,g>=\int_{0}^{1}fgdx$. This inner product defines a metric space structure an subsequently you can have a topological structure.) However, again not sure if you can call this calculus because I don't have an equivalent thing to "differentiation". 
A manifold is a topological space $X$ with a collection of charts which is a homeomorphism $\phi_{\alpha}$ from an open set in $X$ to an open subset of $\mathbb{R}^n$ or $\mathbb{C}$. 
However, given two charts $\phi_{\alpha}:U_{\alpha}\to\mathbb{R}^n$ and $\phi_{\beta}:U_{\beta}\to\mathbb{R}^n$, they need not agree on $U_\alpha\cap U_\beta$. However, $\phi_\beta\circ\phi_\alpha^{-1}:\phi_\alpha(U_\alpha\cap U_\beta)\to\phi_\beta(U_\alpha\cap U_\beta)$ needs to be a suitably well-behaved function. (In the context of Riemann surfaces, that means it needs to be analytic as we are dealing with complex analysis, in other contexts this may mean that it is a continuous or a smooth map from $D\subset\mathbb{R}^n\to\mathbb{R}^n$). 

