# Cantor, longish lines and the Landau -o notations

in general terms this question is about the behaviour of functions of a real variable as their argument $\rightarrow \infty$. i will present the matter as concisely as i can, but my presentation will be rather a confession of ignorance than anything else, and my question, likewise, is more of a request for guidance than a specific puzzle or conjecture. i hope you may forgive a certain prolixity in my attempt to provide a context for and explain the genesis of my puzzlement.

as a person with little formal training in mathematics i am often slow to grasp a new idea, and perhaps as a consequence, i also tend rather often to experience a momentary sensation of utter amazement whenever a penny, or some equivalent piece of mathematical small change, finally drops into the rusty slot-machine of my rational imagination. on one such occasion, some years ago, through the kindness of a faculty member i had borrowed a relatively antique tome by Wacław Sierpinski from the library of the Leeds University Mathematics Department. although the work was far too advanced for my level of understanding, i was nevertheless overwhelmed by what little i was able to apprehend of the contents of this book, devoted to various matters concerning the transfinite ordinals and cardinals. prior to this, i had been familiar with the distinction between countable and uncountable cardinalities, but i had no idea what a fascinating and diverse field of speculative reasoning had been opened up by Cantor's great leap forward. and it interested me greatly when i later read somewhere that the stimulus which originally triggered the direction of Cantor's pioneering research had been something he noticed about the structure of certain point-sets he encountered in the course of his study of some questions involving Fourier expansions. this suggests that the infinite is not always as distant as a naive person would be tempted to assume, and this made it possible for me to conceive, albeit dimly, the thought, which must be a mere commonplace to any decently equipped undergraduate in the present epoch, that the entire family of countable ordinals can be faithfully represented as a subset of a mathematical object as apparently familiar as the interval $[0,1)$ of $\mathbb{R}$.

my present question has to do with a somewhat analogous embedding of the infinite into a more familiar domain, namely, what (i think) Hardy referred to as a classification of real-valued functions by the rapidity with which they grow as their argument becomes very large. of course by a simple transformation of the domain the same phenomena can be observed as the argument approaches some finite point, but i will restrict myself to the paradigmatic case where the argument approaches infinity.

for such discussions the standard technical apparatus is to be found in what are usually called the Landau big-O and little-o symbols. as a matter of practice these symbols are in everyday use in several branches of mathematics, though, despite their undoubted practical utility, i must confess to always having felt they were somewhat deficient in their aesthetic aspect. notwithstanding my idiosyncratic personal aversion, however, these symbols, or some variant on them, are exactly what is needed to define the implied equivalence classes and the total order imposed upon these classes by the growth behaviour of their representative functions at infinity.

at the pre-university level this order is familiar, for example, in the degree of polynomials, or in the fact that the exponential functions grow faster than any positive power function, and that the latter class dominates the logarithms.

what do we see when we look a little more closely?

if we consider$$f_{\alpha}(x)=x^{\alpha} \; (\alpha \in \mathbb{R^+})$$ then it is clear that$$\bar f_{\alpha} \prec \bar f_{\beta}$$ whenever $\alpha \lt \beta$. thus from a topological perspective the classes of power functions form an ensemble which is order-isomorphic to $\mathbb{R^+}$

this whole ensemble is dominated by classes of functions we may denote by $$\bar f_{\alpha,\beta} = \alpha^{\beta x}$$ whose order is equivalent to the lexical ordering on $\mathbb{R}^+ \times \mathbb{R}^+$

this ensemble is again dominated by $$\bar f_{\alpha,\beta,\gamma} = \alpha^{\beta x^{\gamma}} \; (\gamma \gt 1)$$ but dominates $$\bar f_{\alpha,\beta,\gamma} = \alpha^{\beta x^{\gamma}} \; (\gamma \lt 1)$$ it is not difficult to see that there is a whole family of such extensions of the power function, with an order topology isomorphic to the lexical ordering on $(\mathbb{R}^{+})^{\omega}$, but which family is dominated by the "2-parameter" family $$\bar f_{\alpha,\beta} = x^{\alpha x^{\beta}}$$ now, it is not my purpose to continue elaborating these classes, or to discuss the extension of the same scheme in the downward direction beginning with the simple class of logarithms to different bases. i think anyone who has read this far will have a far more developed mathematical imagination than i can ever hope to attain to, and will be correspondingly less bewildered by the baffling scope of the order topology on these function classes, which i suspect, as hinted at in the title, bears some analogy with the long line offered as an interesting example in courses on elementary topology. i would be grateful for any soothing information which might help me to get a better night's sleep. i suspect that the most likely place i should look is recursive function theory, but i do not know much about this, and feel that its confinement to a finitistic domain will not necessarily assist with questions about the nature of the order topology on the full ensemble of function classes.

thank you

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