Ordered lattice of topologies on the 3 point set

[jackferry]

So I made this diagram of the order imposed on the set of topologies on a 3 point set. Each small circle is a topology with the circle elements being the open sets. The larger circles join isomorphic topologies together. The ordering was messy when I tried to draw inclusion from each topology to its relations, so I stuck to drawing arrows from an isomorphic collection to the isomorphic collections its elements belonged to. Any advice, criticisms, feedback? Did I do this correctly? Does the premise make sense? I'm just getting into point set topology, so this very well could be misguided.

 

[robphy]

possibly useful: https://www.math.stonybrook.edu/~olga/mat364-fall14/soln-hw4.pdf

 

[math771]

First of all, great job on creating this diagram! It's clear and visually appealing.

As for your question about whether the premise makes sense, I would say that it does. The idea of ordering topologies on a set based on inclusion and isomorphism is a fundamental concept in point set topology. So, in that sense, your diagram accurately represents this concept.

However, I would suggest adding some labels or annotations to clarify what exactly each circle represents. For example, you could label each circle with the corresponding topology name or list the open sets within each circle. This would make the diagram more informative and easier to understand for someone who is not familiar with point set topology.

Overall, great job on the diagram and keep up the good work in your studies of point set topology!