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 ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}}$

${\displaystyle \bigcup _{i\in \emptyset }M_{i}=\emptyset }$

A set is called a basis of topology if every open set can be written as a union of any number of sets . ${\displaystyle {\mathcal {B}}\subset {\mathcal {O}}}$ ${\displaystyle O}$ ${\displaystyle {\mathcal {B}}}$

Examples

The topology itself forms a basis for any topological space ${\displaystyle (X,{\mathcal {O}})}$

${\displaystyle {\mathcal {B}}_{1}={\mathcal {O}}}$.

For the trivial topology is ${\displaystyle {\mathcal {O}}:=\{X,\emptyset \}}$

${\displaystyle {\mathcal {B}}_{2}:=\{X\}}$

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 : ${\displaystyle {\mathcal {O}}:={\mathcal {P}}(X)}$

${\displaystyle {\mathcal {B}}_{3}:=\{\{x\}\mid x\in X\}}$

The natural topology on owns (by definition) the base ${\displaystyle \mathbb {R} }$

${\displaystyle {\mathcal {B}}_{4}=\{(a,b)\subset \mathbb {R} \mid a,b\in \mathbb {R} \}}$.

The natural topology is also based on a metric space (by definition) ${\displaystyle (X,d)}$

${\displaystyle {\mathcal {B}}_{5}=\{B_{r}(x)\mid r>0,\;x\in X\}}$.

Here is

${\displaystyle B_{r}(x)=\{y\in X\mid d(y,x)<r\}}$

the open sphere around with radius . ${\displaystyle x}$ ${\displaystyle r}$

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 . ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {U}}(x,{\mathcal {O}})}$ ${\displaystyle (X,{\mathcal {O}})}$

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 . ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\mathcal {O}}_{1}}$ ${\displaystyle {\mathcal {O}}_{2}}$ ${\displaystyle {\mathcal {O}}_{1}={\mathcal {O}}_{2}}$

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: ${\displaystyle {\mathcal {M}}}$ ${\displaystyle X}$

• The union of all sets from is equal to the set . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle X}$
• Every intersection of two sets of can be written out as the union of any number of sets . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {M}}}$

Then the basis is a clearly defined topology . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle X}$

The open sets in the topology generated in this way are then precisely those sets that can be represented as a union of sets . ${\displaystyle {\mathcal {M}}}$

Remarks

• Every topological basis of is a sub-basis of , so the basic term intensifies the term sub-basis. ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle (X,{\mathcal {O}})}$
• 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: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {C}}}$

• For every closed set and every out there exists such that and . ${\displaystyle A}$ ${\displaystyle x}$ ${\displaystyle X\setminus A}$ ${\displaystyle C\in {\mathcal {C}}}$ ${\displaystyle A\subset C}$ ${\displaystyle x\notin C}$
• Every union of two sets can be represented as an intersection of sets and it applies . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \bigcap _{C\in {\mathcal {C}}}C=\emptyset }$

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 .