Non-Trivial Discrete Space is not Arc-Connected

Corollary to Non-Trivial Discrete Space is not Connected

Let $T = \struct {S, \tau}$ be a non-trivial discrete topological space.


$T$ is not arc-connected.


Proof

Aiming for a contradiction, suppose $T$ is arc-connected.

From:

Arc-Connected Space is Path-Connected
Path-Connected Space is Connected

we have that $T$ is connected.

But this directly contradicts Non-Trivial Discrete Space is not Connected.

The result follows from Proof by Contradiction.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $10$
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