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.
Read more about this topic: Cylinder Set
Famous quotes containing the words general and/or definition:
“It is a maxim among these lawyers, that whatever hath been done before, may legally be done again: and therefore they take special care to record all the decisions formerly made against common justice and the general reason of mankind.”
—Jonathan Swift (1667–1745)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)