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 closed unit interval $\closedint 0 1$ 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 $\closedint 0 1$.
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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness