Non-Trivial Discrete Space is not Arc-Connected

From ProofWiki
Jump to navigation Jump to search

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