Discrete subset

In mathematics is a room discreetly , if for every point environments are so that no other point is in the area. The points are clearly isolated in space.

Subsets of Euclidean space

Discrete subsets of the real numbers

A subset of the real numbers is said to be discrete if there is an open interval for each element that contains no other element of . The elements of a discrete set are clearly isolated and separated from one another. ${\ displaystyle M \ subset I}$ ${\ displaystyle x \ in M}$ ${\ displaystyle x}$ ${\ displaystyle M}$

For example, the set of integers is a discrete subset of the real numbers. The rational numbers , however, are not discrete, because z. B. for the number 0 there is no open interval that contains no other fractions apart from 0.

Discretion does not mean that there must only be a finite number of elements between every two elements of a discrete set. For example, the set is a discrete subset: for each element there is the open interval that only contains from; the same applies to the elements . However, between and there are an infinite number of elements of . ${\ displaystyle M: = \ {- 1, -1 / 2, -1 / 3, -1 / 4, \ dotsc \} \ cup \ {1.1 / 2.1 / 3.1 / 4, \ dotsc \}}$ ${\ displaystyle 1 / n}$ ${\ displaystyle {] 1 / (n + 1), 1 / (n-1) [}}$ ${\ displaystyle M}$ ${\ displaystyle 1 / n}$ ${\ displaystyle -1 / n}$ ${\ displaystyle -1}$ ${\ displaystyle 1}$ ${\ displaystyle M}$

On the other hand, the set is not discrete because the element 0 is not isolated. ${\ displaystyle M \ cup \ {0 \}}$

Discrete subsets in higher dimensions

Similarly, it is said to be discrete if there is an open environment in for everyone , which does not contain any other element of . The requirement that there is no accumulation point is equivalent . ${\ displaystyle M \ subset \ mathbb {R} ^ {n}}$ ${\ displaystyle x \ in M}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle x}$ ${\ displaystyle M}$ ${\ displaystyle M}$

Discrete metric space

A metric space , the metric of which has the shape for , is called a discrete metric space . ${\ displaystyle d (x, y) = 1}$ ${\ displaystyle x \ neq y}$

properties

A discrete metric space is complete and also discrete as a topological space.

A metric space that is discrete as a topological space, however, does not have to have the discrete metric, nor does it have to be complete. For example, the set given in the section “Discrete Subset of Real Numbers” is a discrete topological space, but the limit value 0 of the Cauchy sequence is outside of . ${\ displaystyle M = \ {- 1 / n, 1 / n \ mid n \ in \ mathbb {N} \}}$ ${\ displaystyle (1.1 / 2.1 / 3, \ dotsc)}$ ${\ displaystyle M}$

Discrete topological space

The concept of the isolated point is generalized to topological spaces by the following definition:

A point in topological space is called an isolated point if the one-element set is open . ${\ displaystyle x}$ ${\ displaystyle X}$ ${\ displaystyle \ {x \}}$

So an isolated point has an environment "in which it is alone". With this term one now generalizes the term discrete subset:

definition

A topological space is called discrete topological space if each of its points is isolated.

properties

Every subset is open in a discrete topological space.
A function in a topological space whose image set is discrete is continuous if and only if it is locally constant .
Every function whose domain is discrete is continuous.

literature

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