Pseudo-random defined class (April 2020)
Since the defined class of numbers is a topology but does not provide variety/non-uniformity (see Flatland defined class variety problem.), what about randomly selecting points? Actual randomness remains undefined, but we can use something like pi to generate which points count, not counting any number a second time.
For instance pi = 11.001001000011111101 in base 2. If we have a small neighborhood, we might say that having it empty of random-points would require an infinite set of zeros found in pi’s formula. But that would make pi rational, which isn’t the case. We can rearrange the numbers with the same class to make this work. For instance, we can have an order to the numbers and count them for a 1 and not for a 0. In this way we can have a topology.
How do we rearrange all the numbers? What we do is, assuming we have n points that we’ve counted and have tested m points next to each on the last turn closer and closer to see if they are in a small enough neighborhood, on the next turn we count m+1 points closer to each of the n points at a time (say half as close every time) and then add an n+1th point. We can eventually fill the whole space and every neighborhood. If the n+1th point is picked to be erased we can still make neighborhoods about it.
In this way, for there to not be any points in any small enough neighborhood we will eventually require infinity point fails in a row, making pi a rational number. Therefore it is a topology.
We know it has variety/non-uniformity because one point might have one point beside it one way but not in the opposite direction!
A discrete topology with variety satisfies our requirements.