I've been working through the proof of de Rham's theorem and as part of this effort, the fact that for a differentiable manifold $M$ there's an isomorphism between the regular and smooth singular homologies of $M$.
This strikes me as an analytic, approximation-type result in spirit: smooth 1-chains are, modulo coefficients, just piece-wise differentiable curves in $M$, so at some level we're just approximating possibly very badly behaved $C^0$ curves in our space by nicer ones. Continuing the 1-dimensional intuition, if I have some nice-enough behaved vector field/differential form-type object, then all I need for a well-defined contour integral is a nice-enough class of curves that I can integrate along, sufficient to recover/reflect the topological structure of my space.
This to me begged the following (perhaps ill-posed) question.
Question: What type of point-set/analytic considerations need be placed on a subset $X$ of $\mathbb{R}^n$ such that we can recover some similar type of result: using only some 'nice enough' (perhaps Lipschitz or something) collection of images of simplices into $X$ and we can recover some theory of integration for something spiritually similar to vector fields/differential forms, ideally reflective of the topology of $X$?
I get that this is all very likely to hinge on the 'nice-enough' handwaving in the above, but I'd be very curious to hear about either examples in this vein or obstructions to defining them in any meaningfully interesting context.
Motivation: Some trivial motivation, say $X \subseteq \mathbb{R}^n$ is convex. If we consider only affine images of simplices into $X$, it would seem intuitive that we recover an analogue that 'every cycle is a boundary' and hence the "homology" of the resulting chain complex is zero, squaring with intuition. Moreover, if you give me any, say "1-cochain," it seems like we'd recover a fairly natural notion of path integral along polygonal paths, complete with a notion of exactness and closedness that work in the intuitive way. Again, this is all meaningless because the collection of spaces is wholly uninteresting topologically, but I'd be curious to see how this type of intuition could be extended (maybe even the above construction to something like open sets or closed sets who are the closure of their interior) in ways I may not have encountered.