You said that you'd often like to talk about (small, or maybe infinitesimal) variations of a smooth map. There is a well known (in some circles at least) topology on the space of smooth maps.
It's called the Whitney $C^{\infty}$-topology.
First you define the Whitney $C^k$-topology using the natural projection $\pi : C^{\infty}(A,B) \twoheadrightarrow J^k(A,B),$ where the base space is the k-jet space; which is identified with $\mathbb{R}^p$, for some $p$ depending on $\dim(A)$ and $\dim(B).$$\dim(B),$ and of course on $k.$ As a basis for the $C^k$-topology on $C^{\infty}(A,B)$ we take the preimages of the open sets in $J^k(A,B)$ under the metric topology induced from $\mathbb{R}^p.$
Informally, the metric on the jet space measures the difference between derivatives. For example, let $f : \mathbb{R} \to \mathbb{R}$ and, for some $x \in \mathbb{R},$ consider $j_x^kf \in J^k(\mathbb{R},\mathbb{R}) \cong \mathbb{R}^{k+2}.$ We have
$j_x^kf \mapsto (x,f(x),f'(x),\ldots,f^{(k)}(x)) \in \mathbb{R}^{k+2}.$
If $W^k$ denotes the set of open sets of $C^{\infty}(A,B)$ under the $C^k$-topology, then the Whitney $C^{\infty}$-topology on $C^{\infty}(A,B)$ is defined to be the topology whose basis is $W$, where
$W := \bigcup_{k=0}^{\infty} W^k.$
The Whitney $C^{\infty}$-topology makes $C^{\infty}(A,B)$ into a Baire space.
See pages 42 - 50 of Golubitsky & Guillemin, "Stable Mappings and their Singularities", (1974).