I am trying to prove that an infinite product of discrete spaces may not be discrete. I tried taking the simplest nontrivial discrete space, $X:=\{0,1\}$ with the discrete topology, and tried to find a sequence of $0$s and $1$s in $\displaystyle\prod_{i=1}^\infty X_i$ that can't be produced by taking intersections of preimages of open sets in the factor spaces under the projection maps, but couldn't come up with anything.
Is this a good approach, or is this actually too simple of an example?
How would you go about solving this problem? I'd be interested in hearing the thought process behind the proof as well. Thanks.