User:Deltahedron/sandbox

=Denseness= In computability theory, denseness or density is c classification of sets of natural numbers in terms of their resprective rates of growth.

Definition
For subset A of the natural numbers, let A(x) denote the number of elements of A which are less than or equal to the natural number x. We define a density order on subsets of the natural numbers by


 * $$ A \preceq B \ \Leftrightarrow\ B(x) \le f(A(x)) \ $$

for a recursive function f.

Properties

 * An infinite set is of lower density than the whole set of natural numbers if anf only if it is a hyperimmune set.
 * The density order defines a distributive lattice on equivalence classes of subsets.