# 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)}$

${\ displaystyle {\ mathcal {U}} (x): = \ {M \ subset X \, | \, M {\ text {is around}} x \}}$.

(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}}}$

${\ displaystyle O {\ text {open in}} {\ mathcal {O}} _ {\ mathcal {M}}}$exactly when .${\ displaystyle O \ in {\ mathcal {M}} (x) {\ text {for all}} x \ in O}$

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

### Filter convergence

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}$

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)}$