Closed set

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In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.[1][2] In a topological space, a closed set can be defined as a set which contains all its limit points. In a metric space, a closed set is a set which is closed under the limit operation.

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Equivalent definitions of a closed set

In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit points.

This is not to be confused with a closed manifold.

Properties of closed sets

A closed set contains its own boundary. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than 2.

Any intersection of closed sets is closed (including intersections of infinitely many closed sets), and any union of finitely many closed sets is closed. In particular, the empty set and the whole space are closed. In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X. The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A can be constructed as the intersection of all of these closed supersets.

Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not be closed.

Examples of closed sets

  • The closed interval [a,b] of real numbers is closed. (See Interval (mathematics) for an explanation of the bracket and parenthesis set notation.)
  • The unit interval [0,1] is closed in the metric space real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ 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.

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