下記 Order type より ”The open interval (0, 1) of rationals is order isomorphic to the rationals (since, for example, f(x)= （2x-1）/(1-｜2x-1｜） is a strictly increasing bijection from the former to the latter);” が、面白いね
https://en.wikipedia.org/wiki/Order_type Order type
For example, the set of integers and the set of even integers have the same order type, because the mapping n→ 2n is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them. To these two order types we may add two more: the set of positive integers (which has a least element), and that of negative integers (which has a greatest element). The open interval (0, 1) of rationals is order isomorphic to the rationals (since, for example, f(x)= （2x-1）/(1-｜2x-1｜） is a strictly increasing bijection from the former to the latter); the rationals contained in the half-closed intervals [0,1) and (0,1], and the closed interval [0,1], are three additional order type examples. (引用終り) 以上
