Does there exist a classification of finite topologies?
I define a finite topology as a finite Set $T$ of Sets which respects the following properties:
- $\forall a,b \in T: a \cap b \in T$,
- $\forall a,b \in T: a \cup b \in T$,
- $ \emptyset \in T$,
- $\exists S\in T\ |\ \forall a \in T , a \subseteq S$.
This seems like a natural thing to do in the vein of classifying finite groups, so i'm curious what current research in this area looks like.