Topologies on Set with 3 Elements
Theorem
Let $S = \set {a, b, c}$ be a set with $3$ elements.
Then there exist $29$ possible different topologies on $S$:
| \(\ds \tau_1\) | \(=\) | \(\ds \set {\O, \set {a, b, c} }\) | Indiscrete Topology | |||||||||||
| \(\ds \tau_2\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set c, \set {a, b}, \set {a, c}, \set {b, c}, \set {a, b, c} }\) | Discrete Topology | |||||||||||
| \(\ds \tau_3\) | \(=\) | \(\ds \set {\O, \set {a, b}, S}\) | Included Set Topology, where the included set is $\set {a, b}$ | |||||||||||
| \(\ds \tau_4\) | \(=\) | \(\ds \set {\O, \set {a, c}, S}\) | Included Set Topology, where the included set is $\set {a, c}$ | |||||||||||
| \(\ds \tau_5\) | \(=\) | \(\ds \set {\O, \set {b, c}, S}\) | Included Set Topology, where the included set is $\set {b, c}$ | |||||||||||
| \(\ds \tau_6\) | \(=\) | \(\ds \set {\O, \set a, S}\) | Excluded Set Topology, where the excluded set is $\set {b, c}$ | |||||||||||
| \(\ds \tau_7\) | \(=\) | \(\ds \set {\O, \set b, S}\) | Excluded Set Topology, where the excluded set is $\set {a, c}$ | |||||||||||
| \(\ds \tau_8\) | \(=\) | \(\ds \set {\O, \set c, S}\) | Excluded Set Topology, where the excluded set is $\set {a, b}$ | |||||||||||
| \(\ds \tau_9\) | \(=\) | \(\ds \set {\O, \set a, \set {b, c}, S}\) | Partition Topology, where the partition is $\set {a \mid b, c}$ | |||||||||||
| \(\ds \tau_{10}\) | \(=\) | \(\ds \set {\O, \set b, \set {a, c}, S}\) | Partition Topology, where the partition is $\set {b \mid a, c}$ | |||||||||||
| \(\ds \tau_{11}\) | \(=\) | \(\ds \set {\O, \set c, \set {a, b}, S}\) | Partition Topology, where the partition is $\set {c \mid a, b}$ | |||||||||||
| \(\ds \tau_{12}\) | \(=\) | \(\ds \set {\O, \set a, \set {a, b}, S}\) | Order Topology, where the total ordering is $a \preccurlyeq b \preccurlyeq c$ | |||||||||||
| \(\ds \tau_{13}\) | \(=\) | \(\ds \set {\O, \set b, \set {a, b}, S}\) | Order Topology, where the total ordering is $b \preccurlyeq a \preccurlyeq c$ | |||||||||||
| \(\ds \tau_{14}\) | \(=\) | \(\ds \set {\O, \set a, \set {a, c}, S}\) | Order Topology, where the total ordering is $a \preccurlyeq c \preccurlyeq b$ | |||||||||||
| \(\ds \tau_{15}\) | \(=\) | \(\ds \set {\O, \set c, \set {a, c}, S}\) | Order Topology, where the total ordering is $c \preccurlyeq a \preccurlyeq b$ | |||||||||||
| \(\ds \tau_{16}\) | \(=\) | \(\ds \set {\O, \set b, \set {b, c}, S}\) | Order Topology, where the total ordering is $b \preccurlyeq c \preccurlyeq a$ | |||||||||||
| \(\ds \tau_{17}\) | \(=\) | \(\ds \set {\O, \set c, \set {b, c}, S}\) | Order Topology, where the total ordering is $c \preccurlyeq b \preccurlyeq a$ | |||||||||||
| \(\ds \tau_{18}\) | \(=\) | \(\ds \set {\O, \set a, \set {a, b}, \set {a, c}, S}\) | Particular Point Topology, where the particular point is $a$ | |||||||||||
| \(\ds \tau_{19}\) | \(=\) | \(\ds \set {\O, \set b, \set {a, b}, \set {b, c}, S}\) | Particular Point Topology, where the particular point is $b$ | |||||||||||
| \(\ds \tau_{20}\) | \(=\) | \(\ds \set {\O, \set c, \set {a, c}, \set {b, c}, S}\) | Particular Point Topology, where the particular point is $c$ | |||||||||||
| \(\ds \tau_{21}\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b}, S}\) | Excluded Point Topology, where the excluded point is $c$ | |||||||||||
| \(\ds \tau_{22}\) | \(=\) | \(\ds \set {\O, \set a, \set c, \set {a, c}, S}\) | Excluded Point Topology, where the excluded point is $b$ | |||||||||||
| \(\ds \tau_{23}\) | \(=\) | \(\ds \set {\O, \set b, \set c, \set {b, c}, S}\) | Excluded Point Topology, where the excluded point is $a$ | |||||||||||
| \(\ds \tau_{24}\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b}, \set {a, c}, S}\) | ||||||||||||
| \(\ds \tau_{25}\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b}, \set {b, c}, S}\) | ||||||||||||
| \(\ds \tau_{26}\) | \(=\) | \(\ds \set {\O, \set a, \set c, \set {a, b}, \set {a, c}, S}\) | ||||||||||||
| \(\ds \tau_{27}\) | \(=\) | \(\ds \set {\O, \set a, \set c, \set {a, b}, \set {b, c}, S}\) | ||||||||||||
| \(\ds \tau_{28}\) | \(=\) | \(\ds \set {\O, \set b, \set c, \set {a, b}, \set {b, c}, S}\) | ||||||||||||
| \(\ds \tau_{29}\) | \(=\) | \(\ds \set {\O, \set b, \set c, \set {a, c}, \set {b, c}, S}\) |
The numbering is arbitrary.
Proof
The power set of $S$ is the set:
- $\powerset S = \set {\O, \set a, \set b, \set c, \set {a, b}, \set {a, c}, \set {b, c}, \set {a, b, c} }$
A topology on $S$ is a subset of $\powerset S$.
Thus the set of all topologies on $S$ is a subset of the power set of $\powerset S$.
From Cardinality of Power Set of Finite Set:
- $\card {\powerset {\powerset S} } = 256$
Half of the subsets of $\powerset {\powerset S}$ do not contain $\O$ and so are not topologies on $S$.
Half of the remaining subsets of $\powerset {\powerset S}$ do not contain $S$ itself, and so are also not topologies on $S$.
We are left with $64$ subsets of $\powerset {\powerset S}$ to investigate.
In the following, they will be numbered arbitrarily.
First we have the discrete topology and indiscrete topology on $S$:
| \(\ds \tau_1\) | \(=\) | \(\ds \set {\O, S}\) | ||||||||||||
| \(\ds \tau_2\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set c, \set {a, b}, \set {a, c}, \set {b, c}, S}\) |
$\Box$
From here, we investigate topologies according to the number of its singletons.
Before doing that, note that a topology on $S$ with either $3$ singletons or $3$ doubletons is the discrete topology.
Hence in the below we need only investigate topologies with $0$, $1$ and $2$ singletons and doubletons.
Topologies with no Singletons
Let $\TT_0$ be a topology on $S$ which does not contain a singleton.
Unless $\TT_0$ is the indiscrete topology, $\TT_0$ must therefore contain at least one doubleton.
Aiming for a contradiction, suppose there exist $2$ distinct doubletons $S_1$ and $S_2$ of $\powerset S$ such that $S_1, S_2 \in \TT_0$.
Their intersection $S_1 \cap S_2$ cannot be $\O$ as then there would be at least $4$ distinct elements in $S$.
On the other hand, $S_1 \cap S_2$ cannot have $2$ distinct elements otherwise $S_1 = S_2$.
So $S_1 \cap S_2$ is a singleton.
But because $\TT_0$ be a topology, $S_1 \cap S_2 \in \TT_0$.
But this contradicts our assertion that $\TT_0$ does not contain a singleton.
So for $\TT_0$ to be a topology, $\TT_0$ must contain exactly $1$ doubleton.
Hence we have the included set topologies on $S$:
| \(\ds \tau_3\) | \(=\) | \(\ds \set {\O, \set {a, b}, S}\) | where the included set is $\set {a, b}$ | |||||||||||
| \(\ds \tau_4\) | \(=\) | \(\ds \set {\O, \set {a, c}, S}\) | where the included set is $\set {a, c}$ | |||||||||||
| \(\ds \tau_5\) | \(=\) | \(\ds \set {\O, \set {b, c}, S}\) | where the included set is $\set {b, c}$ |
$\Box$
Topologies with $1$ Singleton
Let $\TT_1$ be a topology on $S$ which contains exactly $1$ singleton $U$.
First we note that by adding just $1$ singleton to the indiscrete topology on $S$ yields the excluded set topologies on $S$:
| \(\ds \tau_6\) | \(=\) | \(\ds \set {\O, \set a, S}\) | where the excluded set is $\set {b, c}$ | |||||||||||
| \(\ds \tau_7\) | \(=\) | \(\ds \set {\O, \set b, S}\) | where the excluded set is $\set {a, c}$ | |||||||||||
| \(\ds \tau_8\) | \(=\) | \(\ds \set {\O, \set c, S}\) | where the excluded set is $\set {a, b}$ |
Now consider topologies $\TT_1$ on $S$ which contain exactly one doubleton.
Let us add one doubleton $D$ to each of these topologies which contains one singleton $U$.
First suppose that $D \cap U = \O$.
Thus we have the partition topologies on $S$:
| \(\ds \tau_9\) | \(=\) | \(\ds \set {\O, \set a, \set {b, c}, S}\) | where the partition is $\set {a \mid b, c}$ | |||||||||||
| \(\ds \tau_{10}\) | \(=\) | \(\ds \set {\O, \set b, \set {a, c}, S}\) | where the partition is $\set {b \mid a, c}$ | |||||||||||
| \(\ds \tau_{11}\) | \(=\) | \(\ds \set {\O, \set c, \set {a, b}, S}\) | where the partition is $\set {c \mid a, b}$ |
Now suppose that $D \cap U \ne \O$.
Thus we find we have the order topologies on $S$:
| \(\ds \tau_{12}\) | \(=\) | \(\ds \set {\O, \set a, \set {a, b}, S}\) | where the total ordering is $a \preccurlyeq b \preccurlyeq c$ | |||||||||||
| \(\ds \tau_{13}\) | \(=\) | \(\ds \set {\O, \set b, \set {a, b}, S}\) | where the total ordering is $b \preccurlyeq a \preccurlyeq c$ | |||||||||||
| \(\ds \tau_{14}\) | \(=\) | \(\ds \set {\O, \set a, \set {a, c}, S}\) | where the total ordering is $a \preccurlyeq c \preccurlyeq b$ | |||||||||||
| \(\ds \tau_{15}\) | \(=\) | \(\ds \set {\O, \set c, \set {a, c}, S}\) | where the total ordering is $c \preccurlyeq a \preccurlyeq b$ | |||||||||||
| \(\ds \tau_{16}\) | \(=\) | \(\ds \set {\O, \set b, \set {b, c}, S}\) | where the total ordering is $b \preccurlyeq c \preccurlyeq a$ | |||||||||||
| \(\ds \tau_{17}\) | \(=\) | \(\ds \set {\O, \set c, \set {b, c}, S}\) | where the total ordering is $c \preccurlyeq b \preccurlyeq a$ |
Now consider topologies $\TT_1$ on $S$ which contain exactly $2$ doubletons $D_1$ and $D_2$ in addition to its one singleton $U$.
First consider the situation where $D_1 \cap D_2 = U$.
Thus we have the particular point topologies on $S$:
| \(\ds \tau_{18}\) | \(=\) | \(\ds \set {\O, \set a, \set {a, b}, \set {a, c}, S}\) | where the particular point is $a$ | |||||||||||
| \(\ds \tau_{19}\) | \(=\) | \(\ds \set {\O, \set b, \set {a, b}, \set {b, c}, S}\) | where the particular point is $b$ | |||||||||||
| \(\ds \tau_{20}\) | \(=\) | \(\ds \set {\O, \set c, \set {a, c}, \set {b, c}, S}\) | where the particular point is $c$ |
Now consider the situation where $D_1 \cap D_2 \ne U$.
Then $D_1 \cap D_2 \in \TT_1$ by definition of topology.
This would mean that $\TT_1$ contains $2$ singletons.
Hence it is not possible for a topology on $S$ with exactly one singleton $U$ to have $2$ doubletons $D_1$ and $D_2$ such that $D_1 \cap D_2 \ne U$.
Thus we have exhausted our list of topologies on $S$ with one singleton.
$\Box$
Topologies with $2$ Singletons
Let $\TT_2$ be a topology on $S$ which contains exactly $2$ singletons: $U_1$ and $U_2$.
We note that $U_1 \cup U_2$ is also in $\TT_2$.
Thus $\TT_2$ contains at least one doubleton.
This fact yields us the excluded point topologies on $S$:
| \(\ds \tau_{21}\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b}, S}\) | where the excluded point is $c$ | |||||||||||
| \(\ds \tau_{22}\) | \(=\) | \(\ds \set {\O, \set a, \set c, \set {a, c}, S}\) | where the excluded point is $b$ | |||||||||||
| \(\ds \tau_{23}\) | \(=\) | \(\ds \set {\O, \set b, \set c, \set {b, c}, S}\) | where the excluded point is $a$ |
Now consider topologies $\TT_2$ on $S$ which contain exactly $2$ doubletons $D_1$ and $D_2$ in addition to its $2$ singletons $U_1$ and $U_2$.
Suppose $D_1 \cap D_2 \notin \set {U_1, U_2}$.
Then $\TT_2$ would contain $3$ singletons.
So we note that $D_1 \cap D_2 = U_1$ or $D_1 \cap D_2 = U_2$.
This yields the following topologies, which do not appear to be instances of a topology of a particular variety:
| \(\ds \tau_{24}\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b}, \set {a, c}, S}\) | ||||||||||||
| \(\ds \tau_{25}\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b}, \set {b, c}, S}\) | ||||||||||||
| \(\ds \tau_{26}\) | \(=\) | \(\ds \set {\O, \set a, \set c, \set {a, b}, \set {a, c}, S}\) | ||||||||||||
| \(\ds \tau_{27}\) | \(=\) | \(\ds \set {\O, \set a, \set c, \set {a, b}, \set {b, c}, S}\) | ||||||||||||
| \(\ds \tau_{28}\) | \(=\) | \(\ds \set {\O, \set b, \set c, \set {a, b}, \set {b, c}, S}\) | ||||||||||||
| \(\ds \tau_{29}\) | \(=\) | \(\ds \set {\O, \set b, \set c, \set {a, c}, \set {b, c}, S}\) |
That exhausts our possible topologies $\TT_2$ on $S$ which contain exactly $2$ singletons.
$\Box$
Thus we have enumerated all the different topologies on $S$, and the total comes to $29$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 2 \ \text {(ii)}$