Distinctions among limit points

Firstly, I would like to briefly consider the different types of limit points and the nature of their differences. A topological space can be defined entirely in terms of its limit points, as the limit points are precisely the points generated by the closure of a set. In a way topological spaces exist to describe limit points. In typical applications, limits are defined in terms of the behavior of sequences as they approach infinity, so these limits are generated by an infinite process. The definition of a topological space, however, allows for limits that are generated by individual elements.

• Special limit points: limit points that are generated by a finite set
• Analytic limit points: limit points that are generated by an infinite set

The special limit points are generated entirely be the specialization preorder of the topology. This is really then about the relationship between order, topology, and the infinite. Topology only gains its own identity as distinguished from order theory when dealing with the limits of infinite sets. Alexandrov topologies are entirely generated by their special limit points and they are determined by their specialization preorder. Analytic limits arise in the purest sense in order topology when dealing with suborders of non-hereditary-discrete total orders (total orders other then $\mathbb{Z}$). These topologies are Frechet because they are generated entirely by analytic limit points, and metric topologies share this property. Hopefully, this clarifies the nature of topology.