A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts, proposed here, includes a wide class of random equivalence classes.
"Brownian local minima, random dense countable sets and random equivalence classes." Electron. J. Probab. 11 162 - 198, 2006. https://doi.org/10.1214/EJP.v11-309