Trivial Topological Space is Indiscrete
Theorem
Let $T = \struct {S, \tau}$ be a trivial topological space.
Then $\tau$ is the indiscrete topology.
Proof
By definition of trivial topological space, $S$ is a singleton.
That is, $S$ is a set containing exactly one element.
Suppose $S = \set x$ for some object $x$.
Then the power set of $S$ is the set:
- $\powerset S = \set {\O, \set x}$
That is:
- $\powerset S = \set {\O, S}$
Let $\tau$ be a topology on $S$.
We have that $\tau$ is a subset of $\powerset S$.
Hence $\tau$ must equal one of the following sets:
- $\tau_1 = \O$
- $\tau_2 = \set \O$
- $\tau_3 = \set S$
or
- $\tau_4 = \set {\O, S}$.
By definition of a topology, $S \in \tau$.
Thus $\tau \ne \tau_1$ and $\tau \ne \tau_2$.
By Empty Set is Element of Topology, also $\O \in \tau$.
Therefore $\tau \ne \tau_3$.
By Indiscrete Topology is Topology, $\tau_4$ is a topology on $S$.
Hence if $S$ is a singleton, the only possible topology on $S$ is the indiscrete topology.
$\blacksquare$