In topology, a closed set is a set whose complement is open. Many topological properties which are defined in terms of open sets (including continuity) can be defined in terms of closed sets as well. In the familiar setting of a metric space, closed sets can be characterized by several equivalent and intuitive properties, one of which is as follows: a closed set is a set which contains all of its boundary points. They can be thought of as generalizations of closed intervals on the real number line.
In all but the last section of this wiki, the setting will be a general metric space Those readers who are not completely comfortable with abstract metric spaces may think of as being where or for concreteness, and the distance function as being the standard Euclidean distance between two points.
A closed set in a metric space is a subset of with the following property: for any point there is a ball around which is disjoint from
Recall that a ball is the set of all points satisfying The definition of an open set makes it clear that this definition is equivalent to the statement that the complement of is open.
An alternative formulation of closedness makes use of the distance function.
Let be a subset of a metric space and let be a point. Then define Here denotes the infimum or greatest lower bound.
Given this definition, the definition of a closed set can be reformulated as follows:
A subset of a metric space is closed if and only if, for any point
Indeed, if there is a ball of radius around which is disjoint from then has to be at least
Another equivalent definition of a closed set is as follows: is closed if and only if it contains all of its boundary points. This follows from the complementary statement about open sets (they contain none of their boundary points), which is proved in the open set wiki. Indeed, the boundary points of are precisely the points which have distance from both and its complement.
Trivial closed sets: The empty set and the entire set are both closed. This is because their complements are open.
Important warning: These two sets are examples of sets that are both closed and open. "Closed" and "open" are not antonyms: it is possible for sets to be both, and it is certainly possible for sets to be neither. For instance, the half-open interval is neither closed nor open.
These properties follow from the corresponding properties for open sets. Note that the union of infinitely many closed sets may not be closed:
Let be the closed interval in Then which is not closed, since it does not contain its boundary point
Limit points: A point in a metric space is a limit point of a subset if for some sequence of points Here are two facts about limit points:
1. A point is a limit point of if and only if every open ball containing it contains at least one point in which is not
2. A subset of a metric space is closed if and only if it contains all its limit points.
If is a limit point of so that there is a sequence converging to it, then any open ball around must contain some (indeed, all but finitely many) of the On the other hand, if any open ball around contains some points of not equal to then construct by taking to be a point in inside Then because for all
If is closed and is a limit point of which is not in then by the above discussion, is some positive number, say But there is a sequence of points in which converges to so infinitely many of them lie in i.e. their distance to is This is a contradiction.
On the other hand, if is a set that contains all its limit points, suppose Then there is some open ball around not meeting by the criterion we just proved in the first half of this theorem. This is the condition for the complement of to be open, so is closed.
Continuity: A function is continuous if and only if is closed, for all closed sets This follows directly from the equivalent criterion for open sets, which is proved in the open sets wiki.
Note that these last two properties give ways to make notions of limit and continuity more abstract, without using the distance function. In abstract topological spaces, limit points are defined by the criterion in 1 above (with "open ball" replaced by "open set"), and a continuous function can be defined to be a function such that preimages of closed sets are closed.
The closure of a set is defined to be the smallest closed set containing Here are some properties, all of which are straightforward to prove:
equals the intersection of all the closed sets containing
is closed if and only if it equals its closure.
If denotes the complement of then where denotes the interior.
is the union of and its boundary.
equals the set of limit points of
equals the set of points such that
The closure of the interval is This also equals the closure of and
What is the closure of the set of rational numbers in (with the Euclidean distance metric)?
Every real number is a limit point of because we can always find a sequence of rational numbers converging to any real number. One way to do this is by truncating decimal expansions: for instance, to show that is a limit point of consider the sequence of rational numbers. This sequence clearly converges to So the closure of inside is