Advantages of locale theory in calculus

Question

I'm considering whether to learn point-set topology or pointless topology.

Are there advantages to using locales in calculus?

Answer 1

I can't think of any advantages of locale theory to calculus. Moreover, I believe locale theory abstracts topology even further away from convergence theory. Yet convergence theory is at the heart of calculus, which is why I also doubt that locale theory can have advantages at all.

For example, it is (implicitly) convergence theory that accounts for the following key notions in calculus.

1. Limits of sequences, which are used to define precisely the real numbers, and more generally to express methods of approximations.
2. Continuity, i.e. preservation of convergence, which is used to define precisely derivatives of functions.
3. Limits of filters, which are used to define precisely definite integrals, by which approximations are made.

Topology is not fundamental to the above. Rather, in the course of studying convergence, one comes across the topological notions of open sets, closed sets, compact sets, and connected sets. It turns out that in the case of basic (finite-dimensional) calculus, the notions of open sets and closed sets (i.e. the topology) is enough to reconstruct the convergence structure, i.e. notions of limits and continuity, and thus also of compactness. This is why topology is sometimes taken as a starting point in more precise discussions of calculus.

For more advanced applications however (i.e. in infinite-dimensional or variational setting), the topology is not enough. In the orthodox approach, this problem is circumvented in a somewhat ad-hoc way (one considers various topologies and notions of weak convergence).

I suspect that a convergence approach, perhaps through so-called pseudotopological spaces, might have advantages over the ad-hoc constructions, and even over a topology-first approach to calculus. However, this is far from orthodox, and I have no recommendations for beginners to take this route.

Finally, locale theory abstracts topology by making points themselves deduced from the structure of open and closed sets, which makes the convergence structure that is central to calculus even further removed.