Environmental system

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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.


Let a topological space and any one be given .

The environment system or filter of is the set of all environments of and is denoted by. So it is


(A set is called a neighborhood of , if there is a set such that it holds)


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

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



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 .
  • For each is trivial .
  • For and , where is true
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 .

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


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

If each set of systems has been specified which fulfill the four points listed above under properties, a topology can be explained as follows:

exactly when .

This topology is clearly defined and has the set systems as the surrounding systems of .

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 .

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.

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