Question

Prove that an infinite product of discrete spaces may not be discrete.

Answer 1

Consider the simplest nontrivial discrete space $X_i=\{ 0,1\}$ with the discrete topology .

Let $X=Π^{\infty}_{i=1} X_i$

Therefore , By Tychonoff's theorem ,

$X$ is compact and also $X$ is infinite.

But We know that the only compact discrete spaces are the finite ones .

Hence , $X$ is not discrete.

(Proved)