Environment (mathematics)

An epsilon neighborhood ( ) around the number , drawn on the number line . ${\displaystyle \varepsilon }$ ${\displaystyle a}$

Environment is a mathematical term from topology that is used in many areas. It is a generalization of the concept of environment from analysis and specifies the colloquial concept of “environment” for mathematical use. ${\displaystyle \varepsilon }$

Mathematical properties related to a certain environment are called local , as opposed to global .

Environments in metric spaces

The crowd is a neighborhood of the point . ${\displaystyle V}$ ${\displaystyle p}$

The rectangle is not an environment for the corner point . ${\displaystyle V}$ ${\displaystyle p}$

definition

In a metric space , the concept of the environment results from the metric : The so-called environments are defined . For every point in space and every positive real number (epsilon) is defined: ${\displaystyle (X,d)}$ ${\displaystyle d}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle \varepsilon }$

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

The so defined -environment of is also called an open -ball around or open ball. A subset of is now just then a neighborhood of the point when an environment of contains. ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$

Equivalently, the concept of environment can also be defined directly in metric spaces without using the concept of an environment : ${\displaystyle \varepsilon }$

A set is called neighborhood of if and only if there is one such that the property is satisfied for all with . ${\displaystyle U\subseteq X}$ ${\displaystyle x\in X}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle y\in X}$ ${\displaystyle d(x,y)<\varepsilon }$ ${\displaystyle y\in U}$

With quantifiers , the situation can also be expressed as follows:

${\displaystyle \exists \varepsilon >0,\forall y\in X:d(x,y)<\varepsilon \Rightarrow y\in U.}$

Examples

• The definition of the metric turns the set of real numbers into a metric space. The neighborhood of a number is the open interval . ${\displaystyle d(x,y):=|x-y|}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle (x-\varepsilon ,\ x+\varepsilon )}$
• The set of complex numbers also becomes metric space. The -surrounding of a number is the open circular disk around the radius . ${\displaystyle \varepsilon }$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle \varepsilon }$
• Somewhat more generally, all -dimensional real vector spaces have a metric through the usual ( Euclidean norm- induced) distance concept. The environments are here dimensional spheres (in the geometric sense) with a radius . This motivates the more general way of speaking of -balls in other metric spaces as well. ${\displaystyle n}$ ${\displaystyle \varepsilon }$ ${\displaystyle n}$ ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon }$
• An important example from real analysis : The space of restricted functions on a real interval becomes a metric space through the supremum norm . The environment of a restricted function on consists of all functions that approximate point by point with a smaller deviation than . Clear: The diagrams of all of these functions lie within a " tube" around the diagram from . ${\displaystyle I}$ ${\displaystyle \varepsilon }$ ${\displaystyle f}$ ${\displaystyle I}$ ${\displaystyle f}$ ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon }$ ${\displaystyle f}$

For example, take the following amount : ${\displaystyle M}$

This set is a neighborhood of because it is a superset of for one : ${\displaystyle M}$ ${\displaystyle a}$ ${\displaystyle ]a-\varepsilon ,a+\varepsilon [}$ ${\displaystyle \varepsilon >0}$

Environments in topological spaces

A topological space is given. To each point belongs the set of its surroundings. These are primarily the open sets that contain as elements. They are called open environments of. In addition, there are all sets that contain an open environment of as a subset. This means around so if there is an open set in then, is, applies to the ${\displaystyle (X,{\mathcal {T}}).}$ ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {U}}(x).}$ ${\displaystyle {\mathcal {O}}\subseteq X,}$ ${\displaystyle x}$ ${\displaystyle x.}$ ${\displaystyle U\subseteq X}$ ${\displaystyle x}$ ${\displaystyle U\subseteq X}$ ${\displaystyle x,}$ ${\displaystyle U\in {\mathcal {U}}(x),}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}\in {\mathcal {T}}}$ ${\displaystyle x\in {\mathcal {O}}\subseteq U.}$

The set of the surroundings of the point forms a filter with regard to the set inclusion , the ' environment filter ' of . ${\displaystyle {\mathcal {U}}(x)}$ ${\displaystyle x}$ ${\displaystyle x}$

A subset of is called a neighborhood base of , or base of if each neighborhood of contains an element of as a subset. The open surroundings of a point always form the basis of its environmental system. Another example is the neighborhoods of a point in a metric space. Likewise in the squares with the center and positive side length. ${\displaystyle {\mathcal {B}}(x)}$ ${\displaystyle {\mathcal {U}}(x)}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {U}}(x),}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {B}}(x)}$ ${\displaystyle \varepsilon }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle x}$

A subset of a topological space is around the amount , if an open set with exists. ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle S\subseteq X}$ ${\displaystyle O}$ ${\displaystyle S\subseteq O\subseteq U}$

properties

The following properties apply to the environments:

1. Is , then applies . (Every neighborhood of a point contains the point.) ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle x\in U}$
2. Is and so is . (Every superset of a neighborhood of a point is again neighborhood of the point.) ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle U\subset U'\subset X}$ ${\displaystyle U'\in {\mathcal {U}}(x)}$
3. Is and , so also applies . (The intersection of two neighborhoods of a point is again the neighborhood of the point. This means that the intersection of a finite set of neighborhoods of a point is again the neighborhood of the point.) ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle V\in {\mathcal {U}}(x)}$ ${\displaystyle U\cap V\in {\mathcal {U}}(x)}$
4. For each there is one , so that applies to each . (The neighborhood of a point can at the same time be the neighborhood of other points contained in it. In general, a neighborhood of a point is not the neighborhood of all points contained in it, but it contains another neighborhood of , so that the neighborhood of all points is in. ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle V\in {\mathcal {U}}(x)}$ ${\displaystyle U\in {\mathcal {U}}(y)}$ ${\displaystyle y\in V}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle V}$ ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle V}$

These four properties are also called Hausdorff's environmental axioms and form the first historical formalization of the concept of topological space.

Conversely, if one assigns a set system that fulfills the above conditions to each point of a set , then there is a clearly determined topology , so that for each point the system is the surrounding system of . For example, the environments defined above in metric spaces fulfill conditions 1 to 4 and thus clearly define a topology on the set : the topology induced by the metric. Different metrics can induce the same concept of environment and thus the same topology. ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {U}}(x)}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {U}}(x)}$ ${\displaystyle x}$ ${\displaystyle M}$

In this case a set is open if and only if it contains a neighborhood of this point with each of its points. (This sentence motivates the use of the word “open” for the mathematical term defined above: Each point takes its nearest neighbors with it into the open set; clearly speaking, none is “on the edge” of the set.)

Dotted environment

definition

A dotted environment of a point arises from an environment by removing the point , that is ${\displaystyle {\dot {U}}(x)}$ ${\displaystyle x}$ ${\displaystyle U(x)\in {\mathcal {U}}(x)}$ ${\displaystyle x}$

${\displaystyle {\dot {U}}(x):=U(x)\setminus \{x\}}$.

Dotted environments play a role in particular when defining the limit value of a function , as well as in function theory when considering path integrals of holomorphic functions.

example

In a metric space , a dotted neighborhood looks like this : ${\displaystyle (M,d)}$ ${\displaystyle \varepsilon }$

${\displaystyle {\dot {U}}_{\varepsilon }\,(x):=\{\,y\in M\mid 0<d\,(x,y)<\varepsilon \,\}}$

Individual evidence

1. ↑ Querenburg: Set theoretical topology. 1979, p. 20.
2. ↑ Harro Heuser : Textbook of Analysis. Part 1. 8th revised edition. Teubner, Stuttgart a. a. 1990, ISBN 3-519-12231-6 , p. 236 (mathematical guides) .

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin a. a. 2001, ISBN 3-540-67790-9 .
• James R. Munkres : Topology. 2nd Edition. Prentice Hall, Upper Saddle River NJ 2000, ISBN 0-13-181629-2 .