Cylinder set

In mathematics, a cylinder set is the natural open set of a product topology. Cylinder sets are particularly useful in providing the base of the natural topology of the product of a countable number of copies of a set. If V is a finite set, then each element of V can be represented by a letter, and the countable product can be represented by the collection of strings of letters.

Consider the cartesian product ${\displaystyle \textstyle X=\prod _{\alpha }X_{\alpha }\,}$ of topological spaces ${\displaystyle X_{\alpha }}$, indexed by some index ${\displaystyle \alpha }$. The canonical projection is the function ${\displaystyle p_{\alpha }:X\to X_{\alpha }}$ that maps every element of the product to its ${\displaystyle \alpha }$ component. Then, given any open set ${\displaystyle U\subset X_{\alpha }}$, the preimage ${\displaystyle p_{\alpha }^{-1}(U)}$ is called an open cylinder. The intersection of a finite number of open cylinders is a cylinder set. The collection of open cylinders form a subbase of the product topology on ${\displaystyle X}$; the collection of all cylinder sets thus form a basis.

...
Wikipedia