Non-Trivial Arc-Connected Space is Uncountable

Definition
Let $T$ be a topological space consisting of more than one point.

Let $T$ be arc-connected.

Then $T$ is uncountable.

Proof
From Closed Interval in Reals is Uncountable, the unit interval $\left[{0 \,.\,.\, 1}\right]$ consists of an uncountable number of elements.

From the definition of arc-connected, every pair of points in $T$ is either end of the image of an injection from that uncountable set.

From Domain of Injection Not Larger than Codomain it follows that $T$ has a cardinality at least as great as $\left[{0 \,.\,.\, 1}\right]$.

Suppose $T$ were countable.

Then from Subset of Countably Infinite Set is Countable it would follow that the image of an arc in $T$ would also be countable.

But as shown above, it is seen to be uncountable.

So $T$ can not be countable, and is therefore uncountable.