ca.analysis and odes - Is there a topology on growth rates of functions?

There is some fascinating work in the subject of cardinal
characteristics of the continuum in set theory that
directly relates to the concept of growth rates of
functions. I believe that it is the ideas in this subject
that are ultimately fundamental to your question. I explain
a little about the general subject of cardinal
characteristics in my answer
here.

Much of the interest of your question is already present
for functions on the natural numbers. The two main orders on such functions that
one considers in cardinal characteristics are

• almost-less-than, where $f lt^ast g$ means that
  $f(n) lt g(n)$ for all $n$ except finitely often, and


• domination, where $f lt g$ means that $f(n) lt g(n)$ for
  all $n$.

A family $F$ of functions is said to be unbounded if there
is no function $g$ that has $flt^ast g$ for all $fin F$. That is, an unbounded family is a family that is not
bounded with respect to $lt^ast$. A family $F$ is
dominating if every function $f$ is dominated by some
function $gin F$. The corresponding cardinal characteristics
of these two types of families are:

• The bounding number $frak{b}$ is the size of the smallest
  unbounded family.


• The dominating number $frak{d}$ is
  the size of the smallest dominating family.

It is easy to see that $frak{b}leqfrak{d}$, simply
because any dominating family is also unbounded. Also, both
$frak{b}$ and $frak{d}$ are at most the continuum $frak{c}$,
the size of the reals. It is not difficult to see that both
of these numbers must be uncountable, since for any
countable family of functions $f_0,f_1, f_2,ldots$, we can build the function $g(k) = sup_{nleq k}f_n(k)+1$, which eventually exceeds any particular
$f_n$. In other words, any countable family of
functions is bounded with respect either to
almost-less-than or with respect to domination. Thus,
$omega_1leqfrak{b},frak{d}leqfrak{c}$.

It follows from these simple observations that if the
Continuum Hypothesis holds, then both the bounding number
and the dominating number are equal to $omega_1$, which under CH is the same as the continuum $frak{c}$.

Now, the amazing thing is that ZFC independence abounds
with these concepts. First, it is relatively consistent
with ZFC that the Continuum Hypothesis fails, and both the
dominating number and the bounding number are as large as
they could possibly be, the continuum itself, so that $frak{b}=frak{d}=frak{c}$.
Second, it is
also consistent that both are strictly intermediate between
$omega_1$ and the continuum $frak{c}$, but still
equal. Next, it is also consistent with $text{ZFC}+negtext{CH}$ that
the bounding number $frak{b}$ is as small as it could be,
namely $omega_1$, but the domintating number is
much larger, with value $frak{c}$. The tools for proving all
these results and many others involve the method of
forcing.

Now, let me get to the part of my answer that directly
relates to the idea of rates-of-growth. A slalom is
defined to be a sequence of natural number pairs
$(a_0,b_0),
(a_1,b_1), ldots$ with $a_nlt b_n$. Each slalom s corresponds to the collection
of functions $f:omegatoomega$ such that $f(n)$ is in the
interval $(a_n,b_n)$ for all but
finitely many $n$. That is, imagine an olympic athlete on
skiis, who must pass through (all but finitely many of) the
slalom posts. An $h$-slalom is a slalom such that
$|b_n-a_n|leq h(n)$.

Thus, a slalom is a growth rate of functions, in a very
precise sense. With suitably chosen (countable collections
of) slaloms, it is possible to express the concept of
growth rate that you mentioned in your question.

The set theory gets quite interesting. For example, a major
question is: how many slaloms suffice to cover all the
functions? This is particularly interesting when one
restricts the size of the slaloms by considering $h$-slaloms.
A fat slalom is a $2^n$-slalom, where the
$n^{rm th}$ interval has size at most $2^n$.

It turns out that this is connected with ideas involving
meagerness, otherwise known as category. For example,
Bartoszynski
proved that every set of reals of size less than $kappa$ is
meager if and only if for every function $h$ and every family
of $h$-slaloms $F$ of size less than $kappa$, there is a
function $g$ eventually missing every slalom in $F$. In other
words, the possibility of a family of fewer than $kappa$
many $h$-slaloms covering all the functions is equivalent to
every set of size less than $kappa$ being meager.

And so on. There is a large amount of work on these and
similar ideas. An article particularly focused on slaloms
would be
this.
And there is a survey article by
Brendle on
cardinal characteristics.