/sci/ - Science & Math

>>11397169
You know in the way that {x,y} is a subset of {x,y,z}, i.e. we may write
{x,y} < {x,y,z}, many of the topological constructions feel like a "cheap" way enabling model theory.
Then again, {x, a} and {y, z} aren't in an ordered relation..

You know the topologies on a finite set are in 1-1 correspondence with the preorders and then if you add more conditions, you naturally get more refined preorders.
E.g. adding the T0 axiom (>"if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other"), then the T0 topologies of finite spaces are exactly the partial orders.

From that perspective, general topologies having all those dualities (e.g. Boolean algebras, i.e. complemented distributive lattice, being isomorphic to fields of sets (Stone's representation theorem for Boolean algebras
)).

If you come from a set theoretical pov (which is of course a bit unothodox for topology aimed at discussing geometry), then the powerset P of a set X is just a means of _generating_ some partial order given a small number of things X. And to speak of a topology (two, three axioms on P(X)) is to give some basic coherence to the order thus induced.

To say that topology relates to algebra and, in turn, the discussion of possible combinations and even computer science, isn't that far out.