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:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“What I call middle-class society is any society that becomes rigidified in predetermined forms, forbidding all evolution, all gains, all progress, all discovery. I call middle-class a closed society in which life has no taste, in which the air is tainted, in which ideas and men are corrupt. And I think that a man who takes a stand against this death is in a sense a revolutionary.”
—Frantz Fanon (1925–1961)
“I think middle-age is the best time, if we can escape the fatty degeneration of the conscience which often sets in at about fifty.”
—W.R. (William Ralph)