Base (topology)
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.
definition
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 .
Examples
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 .
properties
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 .
Remarks
- 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 .
Web links
- MI Voitsekhovskii, MI Kadets: base . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Eric W. Weisstein : Topological Basis . In: MathWorld (English).
literature
- Gerhard Preuss : General Topology. 2nd, corrected edition. Springer, Berlin et al. 1975, ISBN 3-540-07427-9 , pp. 34-41.
- 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 .