TITLE: Product of totally disconnected space is totally disconnected?
QUESTION [12 upvotes]: I read that the cartesian product with the product topology of a family of totally disconnected topological spaces is totally disconnected, too. 
Is that true?
How are the connected components in the cartesian product with the product topology defined?
Didn't find it yet.
Thanks.

REPLY [6 votes]: Suppose $X=\prod X_\alpha$ where each $X_\alpha$ is a totally disconnected space and $A\subseteq X$ is connected.  For each $\alpha$, the image $\pi_\alpha(A)\subseteq X_\alpha$ must also be connected, where $\pi_\alpha:X\to X_\alpha$ is the projection.  Since $X_\alpha$ is totally disconnected, $\pi_\alpha(A)$ must be a singleton, say $\{x_\alpha\}$.  It follows that $A$ is a singleton, since the only possible element of $A$ is the element $x\in X$ whose $\alpha$th coordinate is $x_\alpha$ for each $\alpha$.
Thus any connected subset of $X$ is a singleton, so $X$ is totally disconnected.
More generally, if $X=\prod X_\alpha$ for arbitrary spaces $X_\alpha$, the connected components of $X$ are just the sets of the form $\prod A_\alpha$ where each $A_\alpha$ is a connected component of $X_\alpha$.  Indeed, the argument above shows that any connected subset $A\subset X$ is contained in exactly one such set $\prod A_\alpha$ (namely, let $A_\alpha$ be the component of $X_\alpha$ containing the connected set $\pi_\alpha(A)$).  On the other hand, a product of connected spaces is connected, so sets of the form $\prod A_\alpha$ are indeed connected.  So these are exactly the maximal connected subsets of $X$.