The 29 Topologies on a 3 Element Set

A topological space consists of a set (a collection of objects) S together with a collection T  of subsets of S, which we call “open sets.” This collection of subsets has to satisfy the following properties:

• ∅ (the empty set) and the set S itself must be in T.
• The union of any members in T  must be in T.
• The intersection of finitely many members in T  must be in T.

In this post, I will discuss the topologies on the set consisting of the three distinct elements a, b, and c:

S = {a, b, c}

Although topologies on finite sets aren’t particularly useful in practice, they are a great tool for learning what a topology is and eliminating any misconceptions that one might have.

There are 29 topologies on the three element set S. Let’s look at a few of them.

We have the trivial topology:

T₁ = {∅, {a, b, c}}.

If we throw in just a singleton set (a set consisting of just one element), we get the following three topologies:

 T₂ = {∅, {a}, {a, b, c}}
T₃ = {∅, {b}, {a, b, c}}
T₄ = {∅, {c}, {a, b, c}}

Note that we can’t throw in just two singleton sets. For example, the following collection is not a topology on S:

{∅, {a}, {b}, {a, b, c}}

Do you see the problem? It’s not closed under taking unions:

{a}  and {b} are there, but {a, b} = {a} ∪ {b}  is not!

However, the following  is a topology on S:

T₅= {∅, {a}, {b}, {a, b}, {a, b, c}}

Below is a picture of all 29 topologies on {a, b, c}  (to avoid clutter we left out the names of the elements). A large circle surrounds a, b, and c in all cases because S = {a, b, c}  is in all 29 topologies. Also, it is understood that the empty set is in all these topologies.

I organized these topologies by the number of sets in each topology. The lowest row consists of just the trivial topology. The next row up consists of the topologies with just one additional set (three sets in total because ∅ and S are in every topology), and so on.

The topology {∅, {a}, {a, b, c}} is finer than the topology {∅, {a, b, c}} because {∅, {a, b, c}} ⊆ {∅, {a}, {a, b, c}}. We can also say that {∅, {a, b, c}} is coarser than {∅, {a}, {a, b, c}}. The topologies {∅, {a}, {a, b, c}}  and {∅, {b}, {a, b, c}} are incomparable. Neither one is finer than the other. To help understand the terminology “finer” and “coarser,” we can picture the open sets as a pile of rocks. If we were to smash that pile of rocks (the open sets) with a hammer, the rocks will break into smaller pieces (creating more open sets), and the pile of rocks (the topology) will have been made “finer.”

Below we see a visual representation of three “chains” of topologies. 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.

 

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