Closed set

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 closed manifold.
Sets that are both open and closed are called clopen sets.

Definition

Given a topological space, the following statements are equivalent:

a set is in
is an open subset of ; that is,
is equal to its closure in
contains all of its limit points.
contains all of its boundary points.

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, 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 .
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:
In terms of net convergence, a point is close to a subset if and only if there exists some net 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, 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 subset of which denotes the closure of in indeed, even if is a closed subset of, it is nevertheless still possible for to be a proper subset of However, is a closed subset of if and only if for some 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. 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
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 F_σ sets. These sets need not be closed.

Examples

The closed interval of real numbers is closed.
The unit interval is closed in the metric space of real numbers, and the set of rational numbers between and 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.
In the finite complement topology on a set, the closed sets are precisely the finite subsets of together with itself.
In the discrete topology on a set, every subset of is closed.
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 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