Open set
In the branch of mathematics topology is an open set with a lot of well-defined property (see below ). Clearly a lot is open when its elements are surrounded only by members of this set if no element of the set is in other words on its edge. The complement of an open set is called a closed set. These quantities are characterized in that they contain all their limit points.
A simple example of an open set the interval in the real numbers. Each real number with the property is surrounded only by numbers with the same property: Choose the amount as environment, then these are the numbers between 0 and 1 is why it is called the interval is an open interval. In contrast, the interval is not open, because "right" from the element 1 ( greater than 1) is not an element of the interval longer.
Whether a lot is open or not, depends on the space in which it lies. The rational numbers form an open set in the rational numbers, but not in the real numbers.
It should be noted that there are both quantities, neither completed nor are open, such as the interval, as well as amounts that are both, as the empty set, the. Such quantities which are open and closed at the same time, are referred to as closed or open set abbreviation of the term as a clopen set.
The distinction of open and closed sets can also meet up with the edge of a crowd. Is this completely to the amount to as it is completed. Belongs to the edge all the way to complement of the set, the set is open.
The concept of open set can be defined at different levels of abstraction. We are here by the pictorial Euclidean space over the metric space to the most general context, the topological space.
- 2.1 Definition
- 2.2 Open ball
- 2.3 Examples
- 2.4 properties
- 3.1 Definition
- 4.1 Affairs
- 4.2 consistency
- 4.3 Outstanding Figure
Euclidean space
Definition
Is a subset of the -dimensional Euclidean space, then called open one if the following holds:
Explanation
Note that from the point dependent, that is, for different points are different. Clearly the amount of points, the distance of less than a ball, and only the interior without the surface. They are called therefore an open ball. ( In this ball is the interior of a circle. ) This ball is the issue raised in the introduction around points.
Metric space
Definition
Be a metric space and a subset of. They call then open (with respect to the induced topology) if and only if:
Again, the choice depends on from. The statement is equivalent to the following: The subset described above is called open if each of its points is an interior point.
Open ball
In analogy to the Euclidean space is called the set of points whose distance as is too small, an open ball. Formally, one writes
And call this quantity the open ball in with center and radius real is.
In the open ball of the edge or the shell of the ball is not included: All of the basic amount, which have a distance smaller than the radius of the center belong to the ball. ( Notice the space given in the article Normalized examples that a ball with respect to a metric "spherical " or is not always " circular ". )
The definition of an open set can now be written as:
Be a metric space. Then is called a subset of open, if the following holds:
This definition is a generalization of the definition of Euclidean spaces, for every Euclidean space is a metric space, and for Euclidean spaces agree the definitions match.
Examples
Considering the real numbers of the conventional Euclidean metric, the following examples are open sets:
- The above open interval that all numbers between 0 and 1 only. This interval is also an example of an open ball in.
- Itself is open.
- The empty set is open.
- The set of rational numbers is open, but not open in.
- The interval is not open in the set of all rational numbers with, however, is open in.
In one can imagine as open sets of quantities for which you have omitted the edge.
Properties
Every open ball is an open set. The proof of this is illustrated by the following figure: To the point of the open ball can be found, namely, so that is very. Be seen analogously in this illustration that each completed ball is completed.
The intersection of two open sets is an open set again. ( For the proof one selects a point from the average, there are then two bullets to the point from which the smaller in both sets, so on average, is located. ) From this one can conclude that the average of finitely many open sets open. In contrast, the average number of open sets must be infinite not be open. For example, looking at the intersection of all open intervals, which runs through all natural numbers, the result is the one-element set, which is not open:
The union of any number (including infinitely many ) open sets is open. ( For the proof one again chooses a point from the union, and there is then a sphere around this point, which is located in one of the combined open sets, so also in the union. )
Topological space
The open balls in metric spaces are the simplest examples of environments in the topology. To define open sets in a more general context, one must abandon the concept of the sphere.
In the definition of a topological space is "openness" is a fundamental concept that is explained by its properties.
Definition
If T is a set of subsets of X with the following properties:
Then we call T a topology on X, and the elements of T are called open sets of the topological space (X, T).
This definition is a generalization of the definition of metric spaces: The set T of all open sets of a metric space (X, d) is a topology such that ( X, T) be a topological space.
Use of the concept of open set
Affairs
Each 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, for example, as the union of all open subsets of A. Note that the subsets must be open in X, not only open in A. ( A itself is always open in A. )
Continuity
If two topological spaces X and Y are given, then a mapping is continuous if each inverse image of an open subset of Y is open in X. Instead of demanding that the inverse image of an open subset is open, you can request that the inverse image of a closed subset is complete. This is an equivalent definition for continuity.
Open Figure
The figure, however, is called open picture if the image of every open set is open. However, you can open not substitute completed the word here as opposed to continuity. The figure with is open, but it forms the closed set to off. Using the open mapping can be examined, the inverse of a bijective mapping on continuity. For a bijection if and only open if its inverse map is continuous. A key sentence from the functional analysis of open linear maps is the set of the open mapping.
A mapping is called relatively open if it is an open mapping onto the subspace topology of its image.