Closed set

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

A simple example is in the interval of real numbers (with the default topology produced by the metric ). The complement of the union of two open intervals, ie an open set, so is a closed set. That is why it is called the interval a closed interval. In contrast, the interval is not complete, because the complement is not open.

Whether a set is closed or not, depends on the space in which it lies. The set of rational numbers is 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 consequences with rational sequence elements that converge to a number outside of the rational numbers.

It should be noted that the term " open set " is not the opposite of " closed set " is: There are volumes, not completed yet been made on how the interval, and 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 open set.

The concept of closed set can be defined at different levels of abstraction. Below the vivid Euclidean space, then metric spaces and topological spaces eventually be considered here.

  • 2.1 Definition
  • 2.2 Completed ball
  • 2.3 Examples
  • 2.4 properties

Euclidean space

Definition

If U is a subset of the n-dimensional Euclidean space, then U is called complete if the following holds:

Explanation

Note that the ε from the point x depends, that is, for various points there are different ε. Clearly the amount of the points whose distance is smaller than ε x, 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. )

The set of all points whose distance from a point x is less than or equal to a positive number r, is also a ball, it is called closed ball as it meets the definition of a closed set.

Properties

M is a subset of the completed and a series of elements of M, which converges, the limit value is also M. This feature can alternatively be used to define the closed subsets.

Each closed set U of can be represented as the intersection of countably many open sets. For example, the closed interval [0,1] is the average of the open intervals for all natural numbers n

Metric space

Definition

Be a metric space and a subset of. Then is called complete if the following applies:

Again, the choice depends on from.

This is equivalent with the following property: If a sequence of elements of U that converges in X, then the limit is in U.

Completed ball

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

And call this quantity the closed ball in X with center x and radius r real is > 0

In the closed rim of the ball or the shell of the ball is included: All of the basic quantity y X x have a distance to the center which is less than or equal to r, are spherical. (Note that in the article Norm ( mathematics) examples given that a sphere with respect to a metric not always "spherical " or is " circular ". )

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

Let ( X, d) be a metric space. Then say a subset U of X complete 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 closed sets:

  • The above closed interval, which are all numbers between 0 and 1, inclusive. This interval is also an example of a completed ball in: The center point is 1/2, the radius is 1/2.
  • Itself is complete.
  • The empty set is complete.
  • The set of rational numbers is completed, but not completed.
  • The interval is not closed in (the circle of pi ), the set of all rational numbers, however, is completed.
  • Finite sets are always complete.
  • As a non - trivial example, one can take an open floor amount, eg. On this amount the interval itself is complete, as each set is complete in itself.

In one can imagine closed sets as sets that contain their edge.

Properties

Every closed ball is a closed set. The proof of this is illustrated by the adjacent diagram: For point y2 outside the closed ball ( x, r ) one finds a ε2, namely ε2 = d (x, y2) - r such that B ( y2, ε2 ) entirely outside of ( x, r) is. You can see analog to this illustration that every open ball is open.

The union of two closed sets is a closed set again. From this one can conclude that the union of a finite number of closed sets is closed. However, the union of infinitely many closed sets need not be completed. United we all singleton sets for the resulting amount neither open nor closed.

The average of any number (including an infinite number ) of closed sets is closed.

Topological space

To define closed sets in a more general context, one must abandon the concept of the sphere. It refers instead only on the openness of the complement.

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

Completed cover

For each subset U of a Euclidean metric or topological space, there is always a smallest closed superset of U, which is called closed hull, also seclusion or completion of U. One can construct the closed hull either as intersection of all closed supersets of U or as a set all limits of all convergent networks that lie in U. An analogue characterization by means of filters is possible. Note, however, that it is not enough in general topological spaces to consider only limits of sequences.

The edge of a subset

Let U be a subset of a topological space. Then it is possible to define the edge of U as the average of the closed shell of U with the closed shell of the complement of U ( or alternatively as the completed shell from the inside of the U without U). So a point lies on the boundary of U, if both points of U as well as points of the complement of U lie in any environment. This boundary term is consistent in metric and Euclidean spaces with the intuitive notion of an edge. In a topological space, then in general:

A set U if and only complete when it contains its boundary.

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