Let \(\mathbb{O}\) be the countable set of all infinite rational number sequences which are ultimately but not identically zero, and let \(H\) contain those elements in \(\mathbb{O}\) whose last nonzero element is positive. With respect to \(\mathbb{O}\), define \(H^\complement\). Then \(f(x) = -x\) is a bijection from \(H\) to \(H^\complement\).
Also, \(H\) looks like a spiral. To see that, require the last nonzero element to be the first, then the second, and then the third one, and watch this sequence of subsets twist into a new dimension at each step.
That's really something! A spiral whose reflection in the origin is its complement.
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