# 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)

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 .