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:
“Towards him they bend
With awful reverence prone; and as a God
Extoll him equal to the highest in Heav’n:
Nor fail’d they to express how much they prais’d,
That for the general safety he despis’d
His own: for neither do the Spirits damn’d
Loose all thir vertue; lest bad men should boast
Thir specious deeds on earth, which glory excites,
Or close ambition varnisht o’er with zeal.”
—John Milton (1608–1674)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)