Completed set

In the sub-area of topology of mathematics , a closed set is a subset of a topological space , the complement of which is an open set .

A simple example is the interval in the real numbers (with the standard topology generated by the metric ). The complement of is the union of two open intervals, so an open set, so is a closed set. That is why the interval is called a closed interval . On the other hand, the interval is not closed because the complement is not open. ${\displaystyle [0,1]}$ ${\displaystyle d_{xy}=\left|x-y\right|}$ ${\displaystyle [0,1]}$ ${\displaystyle \textstyle (-\infty ,0)\cup (1,\infty )}$ ${\displaystyle [0,1]}$ ${\displaystyle [0,1]}$ ${\displaystyle (0,1]}$ ${\displaystyle (-\infty ,0]\cup (1,\infty )}$

Whether a lot is complete or not depends on the space in which it is located. The set of rational numbers with forms a closed set in the rational numbers , but not in the real numbers with the standard topology. This follows from the fact that there are sequences with rational terms that converge to a number outside of the rational numbers. ${\displaystyle x}$ ${\displaystyle 0\leq x\leq 1}$

It should be noted that the term “open set” is not the opposite of “closed set”: there are sets that are neither closed nor open, like the interval , and sets that are both, like the empty set. Such sets that are open and closed at the same time are called closed open sets . ${\displaystyle (0,1]}$

The concept of the closed set can be defined on various levels of abstraction. In the following, the descriptive Euclidean space , then metric spaces and finally topological spaces are considered.

Euclidean space

definition

If U is a subset of the n -dimensional Euclidean space , then U is said to be closed if: ${\displaystyle \mathbb {R} ^{n}}$

For each outside of U there is a , so that every point with , also lies outside U. ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle y\in \mathbb {R} ^{n}}$ ${\displaystyle \|x-y\|<\varepsilon }$

Explanation

Note that the ε depends on the point x , i.e. H. for different points there are different ε. The set of points, whose distance from x is smaller than ε, is a sphere , namely only the interior without the surface. It is therefore also called an open sphere . (In this sphere, this sphere is the inside of a circle .) ${\displaystyle \mathbb {R} ^{2}}$

The set of all points whose distance from a point x is less than or equal to a positive number r is also a sphere; it is called a closed sphere because it fulfills the definition of a closed set.

properties

If M is a closed subset of , and a sequence of elements of M , in the converged, then the limit is of also M . Alternatively, this property can be used to define closed subsets of the . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle (x_{n})}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle (x_{n})}$ ${\displaystyle \mathbb {R} ^{n}}$

Every closed set U vom can be represented as the average of countably many open sets. For example, the closed interval [0,1] is the average of the open intervals     for all natural numbers n. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \textstyle \left(-{\frac {1}{n}},1+{\frac {1}{n}}\right)}$

Metric space

definition

Let be a metric space and a subset of . Then we call it complete if: ${\displaystyle (X,d)}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle U}$

For every out there is a real number , so that for every point out : Aus follows that in lies. ${\displaystyle x}$ ${\displaystyle X\setminus U}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle y}$ ${\displaystyle X}$ ${\displaystyle d(x,y)<\varepsilon }$ ${\displaystyle y}$ ${\displaystyle X\setminus U}$

Again, the choice of depends of starting. ${\displaystyle \varepsilon }$ ${\displaystyle x}$

This is equivalent to the following property: If a sequence of elements of U , in the X converges, then the limit is in U . ${\displaystyle (a_{n})}$

Completed sphere

In analogy to Euclidean space, the set of points y whose distance d ( x , y ) to x is less than or equal to ε is called a closed sphere . Formally you write

${\displaystyle {\overline {B}}_{r}(x):=\{\,y\in X\mid d(x,y)\leq r\,\}}$

and calls this set the closed sphere in X with center x and real radius r > 0.

In the closed sphere, the edge or the envelope of the sphere is included: All y of the basic set X that are at a distance from the center x that is less than or equal to r belong to the sphere. (Note the examples given in the article Norm (mathematics) that a sphere is not always "spherical" or "circular" with respect to a metric.)

The definition of a closed set can now be written as follows:

Let ( X , d ) be a metric space. Then a subset U of X is said to be closed if:

${\displaystyle \forall {x\in X\setminus U}:{\exists \varepsilon }>{0}:B_{\varepsilon }(x)\cap U=\emptyset }$

This definition is a generalization of the definition for Euclidean spaces, because every Euclidean space is a metric space, and for Euclidean spaces the definitions are the same.

Examples

If one considers the real numbers with the usual Euclidean metric, the following examples are closed sets: ${\displaystyle \mathbb {R} }$

• The above-mentioned closed interval , that is all numbers between 0 and 1 inclusive. This interval is also an example of a closed sphere in : the center is 1/2, the radius is 1/2. ${\displaystyle [0,1]}$ ${\displaystyle \mathbb {R} }$
• ${\displaystyle \mathbb {R} }$ itself is complete.
• The empty set is complete.
• The set of rational numbers is closed in , but not closed in . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$
• The interval is not closed in ( is the circle number Pi), whereas the set of all rational numbers with is closed in . ${\displaystyle [0,\pi )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \pi }$ ${\displaystyle x}$ ${\displaystyle 0\leq x<\pi }$ ${\displaystyle \mathbb {Q} }$
• Finite sets are always closed.
• As a non-trivial example, one can take an open basic set, e.g. B. . On this set the interval itself is closed, since every set is closed in itself. ${\displaystyle (0,3)}$ ${\displaystyle (0,3)}$

Im one can think of closed sets as sets that contain their boundary. ${\displaystyle \mathbb {R} ^{2}}$

properties

Closed spheres are closed sets

Every closed ball is a closed set. The proof for this is illustrated by the figure on the left: At the point outside the closed sphere one finds a , namely , so that lies completely outside of . Analogously, you can see from this representation that every open sphere is open. ${\displaystyle y_{2}}$ ${\displaystyle {\overline {B}}(x,r)}$ ${\displaystyle \epsilon _{2}}$ ${\displaystyle \epsilon _{2}=d(x,y_{2})-r}$ ${\displaystyle B(y_{2},\epsilon _{2})}$ ${\displaystyle B(x,r)}$

The union of two closed sets is again a closed set. From this one can conclude that the union of finitely many closed sets is closed. However, the union of infinitely many closed sets does not have to be closed. If you combine all one-element sets for , the resulting set is neither open nor closed. ${\displaystyle \textstyle \left\{{\tfrac {1}{a}}\right\}}$ ${\displaystyle a\in \mathbb {N} }$

The intersection of any number of (also infinitely many) closed sets is closed.

Topological space

To define closed sets in an even more general context, one has to drop the concept of the sphere. Instead, one only refers to the openness of the complement.

Is a topological space and a subset of , then is said to be closed if the complement is an open set . ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle X\setminus U}$

This definition is a generalization of the definition for metric spaces.

Completed shell

For every subset of a Euclidean, metric or topological space there is always a smallest closed superset of , this is called closed envelope , also closure or closure of . One can construct the closed hull either as the average of all closed supersets of or as the set of all limit values ​​of all convergent networks that lie in. Analog characterization with the aid of filter convergence is also possible. Note, however, that in general topological spaces it is no longer sufficient to only consider limit values ​​of sequences. ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$

The edge of a subset

Let be a subset of a topological space. Then it is possible to define the edge of as the intersection of the closed envelope of with the closed envelope of the complement of (or alternatively as the closed envelope of without the interior of ). So a point lies on the edge of if there are points from as well as points from the complement of in every neighborhood . In metric and Euclidean spaces, this concept of edge corresponds to the intuitive concept of edge. In a topological space the following generally applies: ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$

A set is closed exactly when it contains its edge. ${\displaystyle U}$

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .