Closed Set - Examples of Closed Sets

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.

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