Base (topology)

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In set- theoretical topology , a basic discipline of mathematics, a basis is a set system of open sets with certain properties. Topological spaces can be easily defined and classified using bases . Thus, topological spaces that have countable bases satisfy the second countability axiom . They can be considered “small” in the topological sense.


A topological space is given , i.e. a set and a system of open sets . The convention applies

A set is called a basis of topology if every open set can be written as a union of any number of sets .


The topology itself forms a basis for any topological space


For the trivial topology is

One Base. This follows from the above convention about union over an empty index set.

The point sets form a basis for the discrete topology :

The natural topology on owns (by definition) the base


The natural topology is also based on a metric space (by definition)


Here is

the open sphere around with radius .


If for each point in the topological space there is an environment basis of open sets, then the union of all these environment bases forms a basis of the topological space .

The basis of a topological space is not clearly determined. This becomes clear at the basis for the discrete topology: Here, on the one hand, the point sets are already sufficient to form a basis. On the other hand, according to the first example, the entire topology forms a basis, in this case the power set. However, this is almost always significantly larger than the set that only contains the point sets.

In contrast to this, the base uniquely defines a topology, that is to say is a base both from and from , so is .

Construction of topologies from a base

The fact that a base uniquely determines the topology can be used to construct topologies. For this one declares a set system that fulfills certain requirements as the basis. The following applies more precisely:

Is an arbitrary system of subsets of such that:
  • The union of all sets from is equal to the set .
  • Every intersection of two sets of can be written out as the union of any number of sets .
Then the basis is a clearly defined topology .

The open sets in the topology generated in this way are then precisely those sets that can be represented as a union of sets .


  • Every topological basis of is a sub-basis of , so the basic term intensifies the term sub-basis.
  • The concept of the topological basis is not to be confused with the basis of a vector space , the former is a set of open sets, the second a set of vectors, in the case of topological vector spaces a set of points. The terms have a parallel insofar as both create the overall structure in a certain sense , but minimality is in no way required for a topological basis.

Basis of the completed sets

Dual to the above basic term, which applies to the open sets, a base for the closed sets can also be defined. A set system is called a basis of closed sets if each closed set of the topology can be written as an intersection of sets . The following two characterizations are equivalent to this:

  • For every closed set and every out there exists such that and .
  • Every union of two sets can be represented as an intersection of sets and it applies .

Bases of the closed sets occur, for example, in the characterization of T3a spaces .

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