Examples of Closed Sets
- The closed interval of real numbers is closed. (See Interval (mathematics) for an explanation of the bracket and parenthesis set notation.)
- The unit interval is closed in the metric space of real numbers, and the set ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but ∩ Q is not closed in the real numbers.
- Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers.
- Some sets are both open and closed and are called clopen sets.
- Half-interval [1, +∞) is closed.
- The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense.
- Singleton points (and thus finite sets) are closed in Hausdorff spaces.
- If X and Y are topological spaces, a function f from X into Y is continuous if and only if preimages of closed sets in Y are closed in X.
Read more about this topic: Closed Set
Famous quotes containing the words examples of, examples, closed and/or sets:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“One man’s observation is another man’s closed book or flight of fancy.”
—Willard Van Orman Quine (b. 1908)
“There is a small steam engine in his brain which not only sets the cerebral mass in motion, but keeps the owner in hot water.”
—Unknown. New York Weekly Mirror (July 5, 1845)