Non-Trivial Discrete Space is not Arc-Connected
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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:
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: $10$