In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. 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 

By definition, a subset  of a topological space  is called  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  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 ).  
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  of  then there  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  be closed in the "larger" surrounding super-space  
If  and if  is  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 sets 

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 

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  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 
 The closed interval  of real numbers is closed. (See  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 T1 spaces and 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 continuous if and only if preimages of closed sets in  are closed in

See also

Notes

References 

  
  
  
  

General topology