Neighbourhood (mathematics)

Environment is a term of mathematics, is generally defined in the topology and is also used in other areas as part of analysis. He is a generalization of the notion of ε - environment of real analysis and formalized the concept of "closeness" between points in a room.

Mathematical properties that are related to a certain area, called locally as opposed to globally.

• 4.1 Definition
• 4.2 Example

Environments in metric spaces

Definition

In a metric space (M, d) is the area term arises from the metric d: One defines the so-called ε - environments. Ε for each point x of the space M and any positive real number is defined:

The so- defined ε - neighborhood of x is also called the open ε - ball around x or open ball. A subset of M is now exactly then a neighborhood of the point x if it contains an ε - neighborhood of x.

Equivalent can be around term in metric spaces directly without using the concept of an ε - environment define:

With quantifiers can also be expressed by the facts:

Examples

• The set of real numbers is determined by the definition of the metrics to a metric space. The ε - neighborhood of a number x is the open interval (x - ε, x ε ).
• The set of complex numbers is just as much a metric space. The ε - neighborhood of a number z is the open disc centered at z of radius.
• More generally bear all n- dimensional real vector spaces by means of the usual ( induced by the Euclidean norm ) notion of distance a metric. The ε - environments are here n- dimensional spheres ( in the geometric sense) of radius. This motivates the more general way of speaking of ε - balls in other metric spaces.
• An important example of real analysis: The space of bounded functions on a real interval I is the supremum norm on a metric space. The ε - neighborhood of a bounded function f on I here consists of all functions which approximate f pointwise with a smaller deviation than ε. Clearly: The charts of all these functions lying around within a " ε - tube " to the graph of f.

Environments in topological spaces

Given a topological space. For a point is called an open set, for which it holds an open neighborhood of. A subset of is called a neighborhood of the point, if there exists an open set with, so if an open neighborhood of contains.

The set of all neighborhoods of a point forms a filter, around filter means. The filter area is a subset of the power set of.

A subset of is called a base of neighborhoods if every neighborhood has of an element of the subset.

A subset of a topological space is called neighborhood of the set, if there exists an open set with.

Properties

The following properties apply to the environments:

These four properties are also called the Hausdorff axioms environment and form the historically first formalization of the concept of topological space.

For it assigns conversely each point x of a set X is a set system satisfying the above conditions, so there exists a unique topology on X such that for every x, the system is the system of neighborhoods of x. Thus, for example, fulfill the above-defined environments in metric spaces the conditions 1 to 4, and thus determine on the quantity M is clearly a topology: the induced by the metric topology. Various metrics can induce the same environmental concept and thus the same topology.

A lot is in this case if and only open if it contains with each of its points a neighborhood of this point. ( This theorem motivates the use of the word " open" to the above-defined mathematical concept: Each point takes its nearest neighbor in the open set with, no one has spoken clearly " on the edge " of the crowd. )

Dotted around

Definition

A dot neighborhood of a point of an environment formed by removing the item, so

Dotted environments play especially in the definition of the limit of a function a role, as in the theory of functions in the consideration of Wegintegralen holomorphic functions.

Example

In a metric space (M, d ) provides a dotted environment as follows: