Let $\{0,1\}^\omega$ be the Cantor space. I know that its elements are infinite sequences of binary digits (e.g. $0,1,0,1,0,...$)
What are the cylinder sets in the Cantor space $\{0,1\}^\omega$? Are they the clopen sets?
If so, what is their exact definition?
Usually the cylinder sets are the basic open subsets, so those open sets of the form $\prod_n O_n$ where all $O_n = \{0,1\}$ except for some $n \in F$, where $F \subset \omega$ is finite, for which $O_n$ we have a proper open subset. In $\{0,1\}$ we only can choose $\{0\}$ or $\{1\}$ for those (if we have a non-empty open set of course).
So these are exactly all infinite sequences that agree with those finitely many choices of $0$ or $1$ on a finite set $F$ of indices. So it's determined by a choice of such a finite set $F \subset \omega$ and a word $w$ from $\{0,1\}$ of length $|F|$.
These cylinder sets are indeed clopen sets, but finite unions of them are also open (and not always cylinder sets), and of course all clopen subsets are finite unions of them.