Cylinder Set - General Definition

General Definition

Consider the cartesian product of topological spaces, indexed by some index . The canonical projection is the function that maps every element of the product to its component. Then, given any open set, the preimage 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 ; the collection of all cylinder sets thus form a basis.

The restriction that the cylinder set be the intersection of a finite number of open cylinders is important; allowing infinite intersections generally results in a finer topology. In this case, the resulting topology is the box topology; cylinder sets are never Hilbert cubes.