Non-Trivial Discrete Space is not Irreducible
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Corollary to Non-Trivial Discrete Space is not Connected
Let $T = \left({S, \tau}\right)$ be a non-trivial discrete topological space.
$T$ is not irreducible.
Proof
Aiming for a contradiction, suppose $T$ is irreducible.
From Irreducible 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.
$\blacksquare$