Alexandrov topology

In mathematics, any preorder can be given the structure of a topological space by declaring any final section (superior set) open. It can be shown that any "fine" topology comes from that due to (pre)order specialization and, between such spaces, a function is continuous if and only if it is monotonic.

In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open; in Alexandrov topologies, the finite constraint is removed.

This answers a good question: whether every intersection (not just finite intersections) of open sets is open. Answer: This topology is from Alexandrov (also spelled Alexandroff), after Pavel Alexandrov, who was the first to study them.

It is important to note that there are no finite topologies, only their specialization preorders!. Which in turn means (by Henkin's immersion theorem) that preorder is the language of first "order" (in a logical sense) of the topology (but this means: the topology is not "first order" (in a logical sense)). Paradigmatic is the Sierpinski space. But the (infinite) limits of these finite spaces are the spectral spaces.