Topologies on Doubleton
Theorem
Let $S = \set {a, b}$ be a doubleton.
Then there exist $4$ possible different topologies on $S$:
| \(\ds \tau_a\) | \(=\) | \(\ds \set {\O, \set {a, b} }\) | Indiscrete Topology | |||||||||||
| \(\ds \tau_b\) | \(=\) | \(\ds \set {\O, \set a, \set {a, b} }\) | Sierpiński Topology | |||||||||||
| \(\ds \tau_c\) | \(=\) | \(\ds \set {\O, \set b, \set {a, b} }\) | Sierpiński Topology | |||||||||||
| \(\ds \tau_d\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b} }\) | Discrete Topology |
Proof
The power set of $S$ is the set:
- $\powerset S = \set {\O, \set a, \set b, \set {a, b} }$
Because all topologies on $S$ are subsets of $\powerset S$, one of the following must hold:
| \(\ds \tau_1\) | \(=\) | \(\ds \O\) | ||||||||||||
| \(\ds \tau_2\) | \(=\) | \(\ds \set \O\) | ||||||||||||
| \(\ds \tau_3\) | \(=\) | \(\ds \set {\set a}\) | ||||||||||||
| \(\ds \tau_4\) | \(=\) | \(\ds \set {\set b}\) | ||||||||||||
| \(\ds \tau_5\) | \(=\) | \(\ds \set {\O, \set a}\) | ||||||||||||
| \(\ds \tau_6\) | \(=\) | \(\ds \set {\O, \set b}\) | ||||||||||||
| \(\ds \tau_7\) | \(=\) | \(\ds \set {\set a, \set b}\) | ||||||||||||
| \(\ds \tau_8\) | \(=\) | \(\ds \set {\O, \set a, \set b}\) | ||||||||||||
| \(\ds \tau_9\) | \(=\) | \(\ds \set {\set a, \set {a, b} }\) | ||||||||||||
| \(\ds \tau_{10}\) | \(=\) | \(\ds \set {\set b, \set {a, b} }\) | ||||||||||||
| \(\ds \tau_{11}\) | \(=\) | \(\ds \set {\set a, \set b, \set {a, b} }\) | ||||||||||||
| \(\ds \tau_{12}\) | \(=\) | \(\ds \set {\set {a, b} }\) | ||||||||||||
| \(\ds \tau_{13}\) | \(=\) | \(\ds \set {\O, \set {a, b} }\) | ||||||||||||
| \(\ds \tau_{14}\) | \(=\) | \(\ds \set {\O, \set a, \set {a, b} }\) | ||||||||||||
| \(\ds \tau_{15}\) | \(=\) | \(\ds \set {\O, \set b, \set {a, b} }\) | ||||||||||||
| \(\ds \tau_{16}\) | \(=\) | \(\ds \set {\O, \set a, \set b, \set {a, b} }\) |
By definition of a topology, $S$ itself must be an element of the topology.
Thus $\tau_1$ up to $\tau_8$ are not topologies on $S$.
By Empty Set is Element of Topology, for $\tau$ to be a topology for $S$, it is necessary that $\O \in \tau$.
Therefore $\tau_9$ up to $\tau_{12}$ are also not topologies on $S$.
By Indiscrete Topology is Topology, $\tau_{13}$ is a topology on $S$.
By Discrete Topology is Topology, $\tau_{16}$ is a topology on $S$.
It is then seen by inspection that $\tau_{14}$ and $\tau_{15}$ are particular point topologies
Indeed, they are Sierpiński topologies.
By Particular Point Topology is Topology, both $\tau_{14}$ and $\tau_{15}$ are topologies.
Hence the result.
The topologies can be assigned arbitrary labels.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Example $4$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 2 \ \text {(i)}$