Here I sought a transversal of the quotient of $[0,1)$ by some irrational map. Really I was hoping for an explicit construction - something tangible which I could really get to grips with.
It turns out the transversal I sought to visualise is a Lebesgue-immeasurable subset of $[0,1)$, similar to $\Bbb R/\Bbb Q$ and the difficulties visualising it are discussed here, with this answer possibly some practical help but this answer suggesting attempts for some explicit construction may be futile.
Until I learnt the above facts, I was hopeful a construction of the transversal existed which would not be entirely different to the classic construction of the Cantor set, which takes an easily-visualisable process and continues it to its limit.
But the whole "Lebesgue-immeasurable" property looks like a considerable fly in the ointment in that regard, since the Cantor set is of zero measure and therefore, I assume, an inherently different animal.
One of the matters covered there and elsewhere is that the axiom of choice is required in order to give the transversal the linear order it requires in order to be constructed over $[0,1)$. Is that something of a death-knell for the idea of an explicit construction?
The quotient I'm working with, without getting into details, doesn't have a meaningful metric as a subspace of $[0,1)$ (as you would expect) but it does appear to have a meaningful topology as a quotient of a subspace of the lower-limit topology. Is there still hope for an explicit construction, and does the topology sound like a reaonable "gadget" to use in the transversal's construction? The topology does seem to be relevant since it gives hope to the idea of defining least upper bounds on chains, from which Zorn's Lemma can follow (thereby bringing the axiom of choice into play).
