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