Topologies on Doubleton

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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


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.