Motivation: The Borsuk-Ulam theorem is stated in terms of the antipodal map $x\mapsto-x$, but self-homeomorphisms of the sphere can turn many different maps into $-$, so the theorem should hold if we replace negation with any such map. (Maybe these are the continuous involutions of degree$1$ or something; Wikipedia claims that any involution works.)
Generalizing the theorem in this way doesn't completely remove its dependence on the algebraic structure of $\Bbb R$, because the sphere itself is defined algebraically as the locus of $|x|=1$. This seems like just a convenience, though, since we could define the sphere in other ways (e.g. recursively by gluing two cones on a lower-dimensional sphere).
It seems like, with some work, we could state and prove versions of Borsuk-Ulam and similar fixed-point theorems using a "purely topological" version of $\Bbb R$ on which we avoid imposing any algebraic structure. This approach has some aesthetic appeal to me, even though I realize the resulting theorems are no more useful than the standard versions.
How much of topology (I guess I mean the "nice" parts: topology of manifolds / algebraic topology?) can be developed in this way? Has it been done? Are there certain proofs and/or definitions that are hard to reformulate?
How and why does the "rigid" algebraic structure of $\Bbb R$ help us do "stretchy" topology?