Ordered lattice of topologies on the 3 point set

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SUMMARY

The discussion centers on the visualization of the ordered lattice of topologies on a three-point set, where each small circle represents a topology and the elements within are the open sets. The author utilized arrows to indicate isomorphic relationships between topologies, opting not to depict inclusion relations due to complexity. Feedback is sought regarding the correctness of this representation and the overall understanding of point set topology. The provided link offers additional context and examples relevant to the topic.

PREREQUISITES
  • Understanding of point set topology concepts
  • Familiarity with isomorphic structures in topology
  • Basic knowledge of lattice theory
  • Ability to interpret mathematical diagrams
NEXT STEPS
  • Study the properties of topologies on finite sets
  • Learn about isomorphisms in topology
  • Explore lattice theory and its applications in topology
  • Review advanced point set topology resources, such as "Topology" by James Munkres
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Students and educators in mathematics, particularly those focusing on topology, as well as researchers interested in the structural properties of topological spaces.

jackferry
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20171213_174538.jpeg


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.
 

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