Appert topology

In general topology, a branch of mathematics, the Appert topology, named for , is a topology on the set } of positive integers.
In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space "X" with the Appert topology is called the Appert space.
Construction.
For a subset "S" of "X", let denote the number of elements of "S" which are less than or equal to "n":
"S" is defined to be open in the Appert topology if either it does not contain 1 or if it has asymptotic density equal to 1, i.e., it satisfies
The empty set is open because it does not contain 1, and the whole set "X" is open since
formula_3 for all "n".
Related topologies.
The Appert topology is closely related to the Fort space topology that arises from giving the set of integers greater than one the discrete topology, and then taking the point 1 as the point at infinity in a one point compactification of the space. The Appert topology is finer than the Fort space topology, as any cofinite subset of "X" has asymptotic density equal to 1.