Say that $$A := \{f : \mathbb{N} \to \mathbb{R}\}/\sim$$ where $f \sim g$ if $f$ and $g$ are asymptotically equivalent. That is, let $A$ be the set of asymptotic equivalence classes of real functions.
What can be said about $A$? For instance:
- Does the set $A$ have an interesting topology? Metric? Poset structure?
- Is there any kind of canonical representative of a given equivalence class? (I assume such a statement would require some extra conditions, since there are obviously many classes of things that grow ridiculously fast.)
- Does the set $A$ have a name? Is it studied anywhere?
Depending on what you mean by asymptotically equivalent, there's a natural order given by $f \le g$ iff $f = O(g)$. See fast-growing hierarchy and Hardy hierarchy for interesting examples of ordinals embedded into this order.