How many topologies exist on 4 points? Any nomenclature?

[Cruikshank]

Just for fun, I tried enumerating the topologies on n points, for small n. I found that if the space X consists of 1 point, there is only one topology, and for n = 2, there are four topologies, although two are "isomorphic" in some sense. For n = 3, I I found 26 topologies, of 7 types. For n = 4, I found 241 topologies of 21 types.

Did I get those right? And is there a standard nomenclature? For convenience I was defining terms like
ubiquitous point--a point in every set of the topology
extra point--a point that only appears in X and no other set of the topology
dependent point--a point a depends on point b if a in O implies b is in O.
minimal neighborhood--the intersection of all open sets containing a given point (I was trying to find a way to determine an entire topology by giving a "basis" of sorts.)

Since I am only considering small finite sets, I merge mutually dependent points into "set points" (equivalence classes of points are the new points, if you will) and therefore I can make any topology into a T-zero topology, and a function from the set-points to N can preserve the information of how many points were merged.

Describing the structures, I know about discrete and indiscrete topologies, so I would refer to a 2D (discrete subtopology with two elements), a four-nested structure, and so forth. 2D,E in my notation means a topology T contains {0, a, b, ab, abc} (a discrete topology with an extra point added.)

I've proved some theorems, but I'm much better at reinventing the wheel than reading about wheels. I can't find anything on this topic under the names I guessed for it. Could someone please let me know where to find information on this? For all I know it's a computer science thing, or abstract algebra. Thanks in advance.

 

[WWGD]

At this level of finite sets, I think this comes down to set theory. For other questions, maybe you need to look into Filters (logic), etc. There may also be some Combinatorics involved.

 

[micromass]

The names you have invented should be easily expressed in terms of this: https://en.wikipedia.org/wiki/Specialization_(pre)order
See also: https://en.wikipedia.org/wiki/Alexandrov_topology

Besides, for $n=3$, there are $29$ topologies of $9$ types. For $n=4$, there are $355$ topologies of $33$ types.

 

[Cruikshank]

Thank you, micromass! I will study those. That looks like exactly what I wanted.
(By the way, I believe my counts are low because I merged points which only appear together. I will reexamine the cases.)