Indiscrete Space is Arc-Connected iff Uncountable

Theorem
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.

Then $T$ is arc-connected $S$ is an uncountable set.

Proof
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space such that $S$ is uncountable.

Let $a, b \in S$.

Consider an injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.

This can always be found because $S$ is itself uncountable.

From Mapping to Indiscrete Space is Continuous, we have that $f$ is ‎continuous.

Thus $T$ is arc-connected.

Now suppose $S$ is an indiscrete topological space which is arc-connected.

Then there exists an injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.

This can only exist if $S$ is uncountable.