Let $A$ be a finite set with the discrete topology and let $X = A^\Bbb N$ be the product space. Let $ \pi_n:X\to A^n $ be the projection map, i.e. $\pi_n(x_1,\dots,x_{n},x_{n+1},\dots) = (x_1,\dots,x_n)$. Each such map induces a partition of $X$ which we denote as $[x]_n=:\pi_n^{-1}(\pi_n(x))$. Such sets $[x]_n$ are the cylinders of the product topology on $X$. As it has been pointed out here, for each $x$ and $n$ the cylinder $[x]_n$ is clopen in $X$ and hence those are finite unions of cylinders.
My question is the following: is it true that clopen subsets of $X$ are exactly finite unions of cylinders - otherwise I am interested in a counterexample. Does the situation change in case $A$ is countable?
I guess this question is also related, but I am not sure whether answers there apply directly here.