# 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.

$\blacksquare$