[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.
, the following statements are equivalent: An alternative characterization of closed sets is available via sequences and nets.
if and only if every limit of every net of elements of
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
is endowed with the subspace topology induced on it by
this terminology allows for a plain English description of closed subsets: In terms of net convergence, a point
if and only if there exists some net (valued) in
is called a topological super-space of
but to not be closed in the "larger" surrounding super-space
for some (or equivalently, for every) topological super-space
Closed sets can also be used to characterize continuous functions: a map
is continuous at a fixed given point
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
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.
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.
is disconnected if there exist disjoint, nonempty, open subsets
is totally disconnected if it has an open basis consisting of closed sets.
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
have the properties listed above, then there exists a unique topology
The intersection property also allows one to define the closure of a set
which is defined as the smallest closed subset of
can be constructed as the intersection of all of these closed supersets.