# How many topologies exist on 4 points? Any nomenclature?

• Cruikshank
In summary, there are a total of ##29## topologies of ##9## types for ##n=3##, and a total of ##355## topologies of ##33## types for ##n=4##.
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.

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:
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.)

## 1. How many unique topologies can be formed with 4 points?

There are 15 unique topologies that can be formed with 4 points. These include the discrete, indiscrete, line, point, plane, and 10 other topologies.

## 2. What is the nomenclature used for topologies on 4 points?

The nomenclature used for topologies on 4 points is based on the number of open sets in the topology. For example, a topology with 4 open sets is called a T4 topology.

## 3. How many different T4 topologies can be formed with 4 points?

There are 5 different T4 topologies that can be formed with 4 points. These include the discrete, indiscrete, line, point, and plane topologies.

## 4. What is the difference between a T4 topology and a T3 topology on 4 points?

The main difference between a T4 topology and a T3 topology on 4 points is the number of open sets. A T4 topology has 4 open sets, while a T3 topology has only 3 open sets. This means that there are more possible topologies for T4 compared to T3.

## 5. Can a T4 topology on 4 points be homeomorphic to a T3 topology?

No, a T4 topology on 4 points cannot be homeomorphic to a T3 topology. Homeomorphism preserves the number of open sets, so if two topologies have a different number of open sets, they cannot be homeomorphic.

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