Order theory and mathematical analysis

Asymptotic analysis describes the limiting behavior of a function as it approaches infinity. Using asymptotic analysis, we have notions of different rates of growth, so for example, we can have linear, polynomial, or exponential growth rates among others. We can extend this notion to totally order functions by their growth rate, so that growth rates are roughly totally ordered. Though this totally ordered structure is not going to be an ordinary one.

Through the use of order theory, we can extend the real numbers to get non-Archimedean ordered fields that contain both infinite and infinitesimal elements. This includes ordered fields such as the surreal numbers as well as the transseries which characterize growth rates. It has even been shown that the field of transseries carries the structure of the surreal numbers. This shows that the growth rates of functions form the most general order worth studying. This directly applies order theory to asymptotic analysis.

Asymptotic analysis is directly related to the calculus because the derivatives of functions are really another way of looking at the growth rate. So really by looking at order theory, we can see how it is directly applicable to most of mathematical analysis already.

Metric spaces are already explicitly defined in a way that relates them to order theory, as they use an order to relate their distances to one another. Something must be closer to something else in an ordered fashion. Recently, I demonstrated that metric spaces are directly related to the order topology of their set of distances. The order topology of the set of distances places a limit on the topology of the metric space. This demonstrates that really practically all of mathematical analysis can benefit from the use of order theory.

But the relationship goes back in the other direction too because of the order topology. Given any partially ordered set we can explore the topological properties of the order using the open sets formed from it. This shows that topological notions like isolated points can be applied to any partial order. This demonstrates the analytic and topological nature of order theory itself.

Really, metric spaces, orders, and so on are all united by the common thread of topology. Really, it can be said that the common thread at the root of different aspects of mathematical analysis is topology. Orders have a topology on them, and metrics have a topology on themselves and their distance order. As it happens, I consider orders and metrics to be two of the most important structures so thinking about them, the relationship between them, and relating that back to topology has consumed much of my attention.