Cantor ternary set, with removal of middle third reiterated ad infinitum, results in a Cantor dust i.e. ‘ghosts of departed quantities’; that is the rationals (and hence primes) are discarded supposedly, but only for open sets, not end elements. Would such Cantor dust be nowhere dense; and construed as measure zero?
Since the primes are clustered and also rational, therefore Cantor dust would seem clustered. And the non-dust context i.e. complement of dust, would seem to represent the irrationals.
If coarser to finer scale is mapped to rationals, then such local coarser to finer scale is also clustered. Consistent with heterogeneity of nucleosynthesis of Big Bang etc.?
… dust to dust
… ashes to ashes
what is the complement, little circle?