Environmental system

A surrounding system is a special set system in set theoretical topology , a basic discipline of mathematics. A system surrounding a point consists of all sets in which the point is “really contained”, ie is inside . Thus the system of surroundings of a point is the set of all surroundings of a point. Surrounding systems play an important role in topology , where they generalize the concept of convergence for sequences to fit topological spaces . In this context, environmental systems are also called environmental filters.

definition

Let a topological space and any one be given . ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle x\in X}$

The environment system or filter of is the set of all environments of and is denoted by. So it is ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {U}}(x)}$

${\displaystyle {\mathcal {U}}(x):=\{M\subset X\,|\,M{\text{ ist Umgebung von }}x\}}$.

(A set is called a neighborhood of , if there is a set such that it holds) ${\displaystyle M\subset X}$ ${\displaystyle x}$ ${\displaystyle O\in {\mathcal {O}}}$ ${\displaystyle x\in O\subset M}$

example

A set is given , provided with the discrete topology , i.e. every subset of is an open set. Then every amount that contains is always open and thus an environment. So the surrounding system is ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle x}$

${\displaystyle {\mathcal {U}}(x)=\{M\subset X|x\in M\}}$

Conversely , if one considers the indiscrete topology , in which only the entire set and the empty set are open, then the only neighborhood of each point is and thus ${\displaystyle X}$

${\displaystyle {\mathcal {U}}(x)=\{X\}}$.

properties

Environmental systems have the following properties:

• Is and so is . Because if there is an environment of , one exists . But then there is also and thus there is also an environment of . ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle U\subset V}$ ${\displaystyle V\in {\mathcal {U}}(x)}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {O}}\ni O\subset U}$ ${\displaystyle O\subset V}$ ${\displaystyle V}$ ${\displaystyle x}$
• For each is trivial . ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle x\in U}$
• For and , where is true ${\displaystyle I=\{1,\dots ,n\}}$ ${\displaystyle U_{i}\in {\mathcal {U}}(x)}$ ${\displaystyle i\in I}$

${\displaystyle \bigcap _{i\in I}U_{i}:=U_{I}\in {\mathcal {U}}(x)}$

Finite sections of surroundings are thus surroundings again. This follows directly from the cutting stability of the open sets contained in the surroundings.

• For every environment there is an environment , so that an environment is the crowd . ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle V\in {\mathcal {U}}(x)}$ ${\displaystyle U}$ ${\displaystyle V}$

The environmental system is thus a quantity filter , on which the designation as an environmental filter is based.

use

Generation of topologies

Topologies can be defined using environmental systems. To do this, one takes advantage of the fact that a set is open exactly when it is around each of its points. This equates to everyone ${\displaystyle O\in {\mathcal {U}}(x)}$ ${\displaystyle x\in O}$

If each set of systems has been specified which fulfill the four points listed above under properties, a topology can be explained as follows: ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {M}}(x)}$ ${\displaystyle {\mathcal {O}}_{\mathcal {M}}}$

${\displaystyle O{\text{ offen in }}{\mathcal {O}}_{\mathcal {M}}}$exactly when . ${\displaystyle O\in {\mathcal {M}}(x){\text{ für alle }}x\in O}$

This topology is clearly defined and has the set systems as the surrounding systems of . ${\displaystyle {\mathcal {M}}(x)}$ ${\displaystyle x}$

Filter convergence

→ Main article : Filter convergence

In general topological spaces, the usual concept of convergence using sequences is no longer sufficient, so networks or set filters are used to expand the convergence in a meaningful way. A filter is then called convergent to if is. With this new concept of convergence, many formulations for sequences from metric spaces can be formulated equivalently: This is exactly what happens if a filter exists that converges against and contains. Hausdorff spaces can also be characterized using the convergence of filters . ${\displaystyle {\mathcal {F}}}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {F}}\supset {\mathcal {U}}(x)}$ ${\displaystyle x\in {\overline {A}}}$ ${\displaystyle x}$ ${\displaystyle A}$

Additional terms

A lot is a neighborhood basis , if any amount one contains. The thickness of surrounding bases has far-reaching structural consequences. Topological spaces in which all points have countable neighborhood bases are also said to satisfy the first countability axiom . In them, for example, the filter convergence can be dispensed with, the sequence convergence is valid without restriction. ${\displaystyle {\mathcal {B}}(x)\subset {\mathcal {U}}(x)}$ ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle B\in {\mathcal {B}}(x)}$

Web links

• BA Pasynkov: Neighborhood . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .