How many topologies on a 3 point set

As each path moves from the bottom to the top of the picture, we move from coarser to finer topologies. Although there are 29 distinct topologies on a 3-element set, many of them are topologically equivalent. Informally, two topological spaces are topologically equivalent if there is a one-to-one correspondence between the elements of the two spaces that also provides a correspondence between their open sets.

Below is a picture of just 9 of the topologies we have seen. Each of the 29 topologies pictured above is topologically equivalent to one of these 9, and none of these 9 are topologically equivalent to each other.

If you want to begin learning some more advanced mathematics, you may want to check out either Pure Mathematics for Beginners or Topology for Beginners. These books contain no prerequisites and are perfect for anyone just starting out in theoretical math. Join the Newsletter Learn strategies to help you get a perfect score and into your dream school with the Get Newsletter.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. Thanks for the help. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. Topologies on a 3 point space Ask Question. Asked 8 years, 8 months ago. Active 6 years, 7 months ago. Viewed 2k times. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself.

This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.

SE site.