Everything about Closed Set totally explained
In
topology and related branches of
mathematics, a
closed set is a
set whose
complement is
open.
Definition 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 isn't to be confused with a
closed manifold.
Properties of closed sets
A closed set contains its own
boundary. In other words, if you're "outside" a closed set and you "wiggle" a little bit, you'll stay outside the set. Note that this is also true if the boundary is the empty set, for example in the metric space of rational numbers, for the set of numbers of which the square is less than 2.
Any
intersection of arbitrarily many closed sets is closed, 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's 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 intervals 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 isn't 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.
More about closed sets
In
point set topology a set
is closed if it contains all its
boundary points.
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.
An alternative characterization of closed sets is available via
sequences and
nets.
A subset
A of a topological space
X is closed in
X if and only if every
limit of every net of elements of
A also belongs to
A.
In a
first-countable space (such as a metric space), it's enough to consider only
sequences, instead of all nets.
One value of this characterisation 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 characterisation also depends on the surrounding space
X, because whether or not a sequence or net converges in
X depends on what points are present in
X.
We have seen twice that whether a set is closed is relative; it depends on the space that it's embedded in. However, the
compact Hausdorff spaces are "absolutely closed" in a certain sense.To be precise, if you embed a compact Hausdorff space
K in an arbitrary Hausdorff space
X, then
K will always be a closed subset of
X; the "surrounding space" doesn't matter here. In fact, this property characterizes the compact Hausdorff spaces.
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.
Further Information
Get more info on 'Closed Set'.
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