How many topologies exist on 4 points? Any nomenclature?

  • #1
82
4
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.
 

Answers and Replies

  • #2
WWGD
Science Advisor
Gold Member
2019 Award
5,341
3,295
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.
 
Last edited:
  • #4
82
4
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.)
 

Related Threads on How many topologies exist on 4 points? Any nomenclature?

Replies
2
Views
3K
  • Last Post
Replies
4
Views
3K
Replies
5
Views
2K
Replies
2
Views
4K
  • Last Post
Replies
8
Views
4K
Replies
1
Views
4K
Replies
1
Views
854
Replies
18
Views
3K
Replies
38
Views
7K
Replies
6
Views
1K
Top