Differentiable maps between topological spaces [closed]

Question

Is it possible to define differentiable maps between topological spaces without using the idea of manifolds? I mean with using just the topological structure (open sets or neighborhoods).

Answer 1

While this is not for general topological spaces, you might be interested in a kind of generalization of differential calculus for metric spaces, developed by Burroni and Penon here.

A key notion is that of two functions $f, g: M \to N$ between metric spaces to be tangent to each other at a point $a \in M$: this means $f(a) = g(a)$ and the function $C: M \to \mathbb{R}_+$ defined by

$$C(x) = \frac{d(f(x), g(x))}{d(x, a)} \;\; (x \neq a), \qquad C(a) = 0$$

is continuous at $a$. This is an equivalence relation. Then, given pointed metric spaces $(M, a)$ and $(N, b)$, one defines a jet $(M, a) \to (N, b)$ to be a tangency equivalence class of locally Lipschitz maps $f: M \to N$ such that $f(a) = b$ ("locally Lipschitz" means there is some $k > 0$ such that $f$ is $k$-Lipschitz when restricted to some neighborhood of $a$). If $f: (M, a) \to (N, b)$ is a based function that is tangent to some locally Lipschitz $g: (M, a) \to (N, b)$ at $a$, then they define the tangential (roughly akin to a differential) $T f_a$ to be the corresponding jet, and in that case they say $f$ is "tangentiable" at $a$.

Ultimately Burroni and Penon want to define, given a map $f: M \to N$ between suitable metric spaces, a notion of tangential $tf: M \to {"Jet(M, N)"}$ that plays a role analogous to the differential $df: U \to L(E, E')$ of a differentiable map $f: U \to E'$, where $E, E'$ are normed vector spaces and $U$ is an open set of $E$. They achieve this by considering certain metric spaces equipped with some extra structure, called transmetric spaces. But for this I'll have to refer you to the article.