Indiscrete Space is Arc-Connected iff Uncountable

Theorem

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.


Then $T$ is arc-connected if and only if $S$ is an uncountable set.


Proof

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space such that $S$ is uncountable.

Let $a, b \in S$.


Consider an injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.

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 $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.


This can only exist if $S$ is uncountable.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology: $9$
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