Quasi-Partial Order on Finite Topologies

Random number: 91

22nd January 2017

So I've started thinking about topologies on a finite number of elements. Consider the graph such that the vertices are all the topologies on \(n\) distinct elements, and there is a directed edge from topology \(i\) to topology \(j\) iff there is some subset of the \(n\) elements \(A\) such that \(T_i \cup A\) forms a subbasis for \(T_j\), where \(T_i,T_j\) represent the topologies represented by the vertices at \(i,j\) respectively. This sort of forms a quasi-partial order on the topologies on \(n\) elements (as in, it looks kind of like some sort of ordering because if you draw the graph all the arrows seem to point in a single direction; but I don't think it actually counts as any type of real 'order', so for now its 'quasi-partial'). I'll be back tomorrow with some (hopefully) sik graphs.