Topologies on Doubleton

Theorem
Let $S = \set {a, b}$ be a doubleton.

Then there exist $4$ possible different topologies on $S$:

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:


 * $\tau_1 = \O$
 * $\tau_2 = \set \O$
 * $\tau_3 = \set {\set a}$
 * $\tau_4 = \set {\set b}$
 * $\tau_5 = \set {\O, \set a}$
 * $\tau_6 = \set {\O, \set b}$
 * $\tau_7 = \set {\set a, \set b}$
 * $\tau_8 = \set {\O, \set a, \set b}$
 * $\tau_9 = \set {\set a, \set {a, b} }$
 * $\tau_{10} = \set {\set b, \set {a, b} }$
 * $\tau_{11} = \set {\set a, \set b, \set {a, b} }$
 * $\tau_{12} = \set {\set {a, b} }$
 * $\tau_{13} = \set {\O, \set {a, b} }$
 * $\tau_{14} = \set {\O, \set a, \set {a, b} }$
 * $\tau_{15} = \set {\O, \set b, \set {a, b} }$
 * $\tau_{16} = \set {\O, \set a, \set b, \set {a, b} }$

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.