Open amount

In the sub-area of topology of mathematics , an open set is a set with a well-defined property (see below). A set is clearly open if its elements are only surrounded by elements of this set, in other words if no element of the set lies on its edge . The complementary set of an open set is called a closed set . These sets are characterized by the fact that they all contain their accumulation points .

A simple example of an open set is the interval in the real numbers . Every real number with the property is only surrounded by numbers with the same property: Choose the set as the environment , then these are the numbers between 0 and 1. This is why the interval is called an open interval . On the other hand, the interval is not open, because “to the right” of element 1 (greater than 1) is no longer an element of the interval . ${\displaystyle (0,1)}$ ${\displaystyle x}$ ${\displaystyle 0<x<1}$ ${\displaystyle (x/2,1/2+x/2)}$ ${\displaystyle (0,1)}$ ${\displaystyle (0,1]}$ ${\displaystyle (0,1]}$

Whether a crowd is open or not depends on the space in which it is located. The rational numbers with form an open set in the rational numbers , but not in the real numbers, since every interval of real numbers with more than one element also contains irrational numbers. ${\displaystyle x}$ ${\displaystyle 0<x<1}$

It should be noted that there are both sets that are neither closed nor open, such as the interval , and sets that are both, such as the empty set . Such sets, which are open and closed at the same time, are called closed open sets or, after the English term, clopen sets . ${\displaystyle (0,1]}$

The distinction between open and closed sets can also be made with the help of the boundary of a set. If this completely belongs to the set, it is complete. If the edge belongs completely to the complement of the set, then the set is open.

The concept of the open set can be defined on various levels of abstraction. We are going here from the descriptive Euclidean space via the metric space to the most general context, the topological space .

Euclidean space

definition

If a subset of the -dimensional Euclidean space is called open , if the following applies: ${\displaystyle U}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle U}$

For every out there is a real number , so that every point of the whose distance is too smaller than , lies in. ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle y}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$ ${\displaystyle U}$

Explanation

Note that depends on the point , i.e. that is, for different points there are different ones . The set of points, the distance of which is less than a sphere , is clear , and 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 .) This sphere is the environment of points mentioned in the introduction . ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle U}$

Metric space

definition

Let be a metric space and a subset of . It is then called open (with regard to the topology induced by) if: ${\displaystyle (X,d)}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle d}$

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

Again, the choice of depends of starting. The statement is equivalent to the following: The subset described above is called open if each of its points is an internal point . ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle U}$

Open sphere

In analogy to Euclidean space, the set of points whose distance is too small is called an open sphere . Formally you write ${\displaystyle y}$ ${\displaystyle d(x,y)}$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$

${\displaystyle U_{r}(x):=\{y\in X\mid d(x,y)<r\}}$

and calls this set the open sphere in with a center and a real radius . ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle r>0}$

In the open ball of the edge or the shell of the sphere is not included with: All of the basic amount that the center of a smaller distance than the radius have, belong to the ball. (Note the examples given in the article Normalized Space that a sphere is not always "spherical" or "circular" with respect to a metric.) ${\displaystyle y}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle r}$

The definition of an open set can now be written as follows:

Be a metric space. Then a subset of is called open if: ${\displaystyle (X,d)}$ ${\displaystyle U}$ ${\displaystyle X}$

${\displaystyle \forall \,x\in U\;\exists \,\varepsilon >0\colon \;U_{\varepsilon }(x)\subseteq U}$.

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

Looking at the real numbers with the usual Euclidean metric, the following examples are open sets: ${\displaystyle \mathbb {R} }$

• The above-mentioned open interval , these are all numbers between 0 and 1 only. This interval is also an example of an open ball in . ${\displaystyle (0,1)}$ ${\displaystyle \mathbb {R} }$
• ${\displaystyle \mathbb {R} }$ itself is open.
• The empty set is open.
• The set of rational numbers is open in , but not open in . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$
• The interval is not open in , whereas the set of all rational numbers with is open in . ${\displaystyle (0,\pi ]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle 0<x\leq \pi }$ ${\displaystyle \mathbb {Q} }$

Im one can think of open sets as sets in which the border has been left out. ${\displaystyle \mathbb {R} ^{2}}$

Consider an arbitrary set with the discrete metric ${\displaystyle M}$

${\displaystyle d(x,y):={\begin{cases}0&\mathrm {f{\ddot {u}}r} \ x=y\\1&\mathrm {f{\ddot {u}}r} \ x\neq y,\end{cases}}}$

then every subset is open. In particular, sets that contain only a single point are open. This is easily seen by looking at an open sphere around a . If you choose , you are simply in the vicinity . ${\displaystyle U\subset M}$ ${\displaystyle U_{r}(x)}$ ${\displaystyle x\in U}$ ${\displaystyle r<1}$ ${\displaystyle x}$ ${\displaystyle U_{r}(x)}$

properties

Open spheres are open sets

Every open ball is an open set. The proof of this is illustrated by the following figure: At the point of the open sphere one finds a , namely , so that lies entirely in . Analogously, you can see from this representation that every closed sphere is closed. ${\displaystyle y_{1}}$ ${\displaystyle U(x,r)}$ ${\displaystyle \varepsilon _{1}}$ ${\displaystyle \varepsilon _{1}=r-d(x,y_{1})}$ ${\displaystyle U(y_{1},\varepsilon _{1})}$ ${\displaystyle U(x,r)}$

The intersection of two open sets is again an open set. (To prove it, one chooses a point from the average; there are then two spheres around the point, of which the smaller one lies in both sets, i.e. in the average.) From this one can conclude that the intersection of finitely many open sets is open. In contrast, the intersection of an infinite number of open sets does not have to be open. For example, if you consider the intersection of all open intervals , where all natural numbers run through, the result is the one-element set that is not open: ${\displaystyle \mathbb {R} }$ ${\displaystyle (-{\tfrac {1}{a}},{\tfrac {1}{a}})}$ ${\displaystyle a}$ ${\displaystyle \{0\}}$

${\displaystyle \bigcap _{a\in \mathbb {N} }\left(-{\frac {1}{a}},{\frac {1}{a}}\right)=\left[\lim _{a\to \infty }-{\frac {1}{a}},\lim _{a\to \infty }{\frac {1}{a}}\right]=[0,0]=\{0\}}$

The union of any number (i.e. also an infinite number) of open sets is open. (For the proof one again selects a point from the union; there is then a sphere around this point, which lies in one of the united open sets, i.e. also in the union.)

Topological space

The open spheres in metric spaces are the simplest examples of environments in topology. To define “openness” in an even more general context, one has to drop the concept of the sphere. Open sets, which can only be explained by their properties, are fundamental for the definition of a topological space .

Let be a set of subsets of the given basic set with the following properties: ${\displaystyle {\mathcal {T}}}$ ${\displaystyle X}$

1. The empty set and the basic set are elements of . ${\displaystyle \emptyset }$ ${\displaystyle X}$ ${\displaystyle {\mathcal {T}}}$
2. Every union of elements of is itself an element of . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$
3. The intersection of finitely many elements of is element of . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$

Then called a topology on , and the elements of hot open sets of the topological space . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle (X,{\mathcal {T}})}$

This definition is a generalization of the definition for metric spaces: the set of all open sets of a metric space is a topology, so is a topological space. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle (X,d)}$ ${\displaystyle (X,{\mathcal {T}})}$

Use of the term open set

Discrete topology

The discrete topology can be on any set X define. It is the topology under which all subsets of X are open. It matches the topology induced by the discrete metric mentioned above .

Interior

→ Main article : Inner point

Every subset A of a topological (or metric) space X contains a (possibly empty) open set. The largest open subset of A is called the interior of A ; to get it as the union of all open subsets of A . Note that the open subsets in X must be not only open in A . ( A itself is always open in A. )

continuity

→ Main article : Continuous function

If two topological spaces X and Y given, then a is imaging is continuous, if each archetype an open subset of Y disclosed in X is. Instead of demanding that the archetype of an open subset be open, one can claim that the archetype of a closed subset is closed. That is an equivalent definition for continuity. ${\displaystyle f\colon X\to Y}$

Open figure

→ Main article : Open figure

On the other hand, the mapping is called open mapping if the picture of each open set is open. However, in contrast to continuity, the word open cannot be replaced by closed here. The illustration with is open, but shows the closed set on . With the help of the open mapping one can now examine the inverse of a bijective mapping for continuity. Because a bijective mapping is open if and only if its inverse mapping is continuous. A central theorem from the functional analysis of open linear mappings is the open map principle . ${\displaystyle f\colon X\to Y}$ ${\displaystyle p\colon \mathbb {R} ^{2}\to \mathbb {R} }$ ${\displaystyle (s,t)\mapsto s}$ ${\displaystyle \{(s,t)\colon s\geq 0,st\geq 1\}}$ ${\displaystyle ]0,\infty [}$

A mapping is called relatively open if it is an open mapping to the subspace topology of its image. The complementary concept to the open mapping is the completed mapping .

literature

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