Indiscrete Space is Arc-Connected iff Uncountable

Theorem
Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Then $T$ is arc-connected iff $S$ is an uncountable space.

Proof
Let $S$ be an indiscrete topological space which is uncountable.

Let $a, b \in S$.

Consider an injection $f: \left[{0. . 1}\right] \to X$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

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

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

Thus $S$ is arc-connected.

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

Then there exists an injection $f: \left[{0. . 1}\right] \to X$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

This can only exist if $S$ is uncountable.