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 complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

Equivalent definitions of a closed setEdit

By definition, a subset   of a topological space   is called closed if its complement   is an open subset of  ; that is, if   A set is closed in   if and only if it is equal to its closure in   Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset   is always contained in its (topological) closure in   which is denoted by   that is, if   then   Moreover,   is a closed subset of   if and only if  

An alternative characterization of closed sets is available via sequences and nets. A subset   of a topological space   is closed in   if and only if every limit of every net of elements of   also belongs to   In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets. One value of this characterization is that it may be used as a definition in the context of convergence spaces, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space   because whether or not a sequence or net converges in   depends on what points are present in   A point   in   is said to be close to a subset   if   (or equivalently, if   belongs to the closure of   in the topological subspace   meaning   where   is endowed with the subspace topology induced on it by  [note 1]). Because the closure of   in   is thus the set of all points in   that are close to   this terminology allows for a plain English description of closed subsets:

a subset is closed if and only if it contains every point that is close to it.

In terms of net convergence, a point   is close to a subset   if and only if there exists some net (valued) in   that converges to   If   is a topological subspace of some other topological space   in which case   is called a topological super-space of   then there might exist some point in   that is close to   (although not an element of  ), which is how it is possible for a subset   to be closed in   but to not be closed in the "larger" surrounding super-space   If   and if   is any topological super-space of   then   is always a (potentially proper) subset of   which denotes the closure of   in   indeed, even if   is a closed subset of   (which happens if and only if  ), it is nevertheless still possible for   to be a proper subset of   However,   is a closed subset of   if and only if   for some (or equivalently, for every) topological super-space   of  

Closed sets can also be used to characterize continuous functions: a map   is continuous if and only if   for every subset  ; this can be reworded in plain English as:   is continuous if and only if for every subset     maps points that are close to   to points that are close to   Similarly,   is continuous at a fixed given point   if and only if whenever   is close to a subset   then   is close to  

More about closed setsEdit

The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces.

Whether a set is closed depends on the space in which it is embedded. However, the compact Hausdorff spaces are "absolutely closed", in the sense that, if you embed a compact Hausdorff space   in an arbitrary Hausdorff space   then   will always be a closed subset of  ; the "surrounding space" does not matter here. Stone–Čech compactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.

Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed.

Closed sets also give a useful characterization of compactness: a topological space   is compact if and only if every collection of nonempty closed subsets of   with empty intersection admits a finite subcollection with empty intersection.

A topological space   is disconnected if there exist disjoint, nonempty, open subsets   and   of   whose union is   Furthermore,   is totally disconnected if it has an open basis consisting of closed sets.

Properties of closed setsEdit

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  

  • Any intersection of any family of closed sets is closed (this includes intersections of infinitely many closed sets)
  • The union of finitely many closed sets is closed.
  • The empty set is closed.
  • The whole set is closed.

In fact, if given a set   and a collection   of subsets of   such that the elements of   have the properties listed above, then there exists a unique topology   on   such that the closed subsets of   are exactly those sets that belong to   The intersection property also allows one to define the closure of a set   in a space   which is defined as the smallest closed subset of   that is a superset of   Specifically, the closure of   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 setsEdit

  • 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   of rational numbers between   and   (inclusive) is closed in the space of rational numbers, but   is not closed in the real numbers.
  • Some sets are neither open nor closed, for instance the half-open interval   in the real numbers.
  • Some sets are both open and closed and are called clopen sets.
  • The ray   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.
  • The set of integers   is an infinite and unbounded closed set in the real numbers.
  • If   is a function between topological spaces then   is a continuous if and only if preimages of closed sets in   are closed in  

See alsoEdit

NotesEdit

  1. ^ In particular, whether or not   is close to   depends only on the subspace   and not on the whole surrounding space (e.g.   or any other space containing   as a topological subspace).

ReferencesEdit

  1. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X.
  2. ^ Munkres, James R. (2000). Topology (2nd ed.). [Prentice Hall]]. ISBN 0-13-181629-2.