# 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 if and only if $S$ is an uncountable set.

## Proof

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

Let $a, b \in S$.

Consider an injection $f: \left[{0 \,.\,.\, 1}\right] \to S$ 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 $T$ is arc-connected.

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

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

This can only exist if $S$ is uncountable.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 4: \ 9$