# 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. ${\ displaystyle {\ mathcal {B}} (x) \ subset {\ mathcal {U}} (x)}$ ${\ displaystyle U \ in {\ mathcal {U}} (x)}$ ${\ displaystyle B \ in {\ mathcal {B}} (x)}$ 