# Gδ set

# Gδ set

In the mathematical field of topology, a **Gδ set** is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with *G* for *Gebiet* (*German*: area, or neighbourhood) meaning open set in this case and δ for *Durchschnitt* (*German*: intersection). The term **inner limiting set** is also used. Gδ sets, and their dual Fσ sets, are the second level of the Borel hierarchy.

Definition

`In a topological space a **Gδset** is acountableintersectionofopen sets. The Gδsets are exactly the levelsets of theBorel hierarchy.`

Examples

Any open set is trivially a Gδ set

The irrational numbers are a Gδ set in the real numbers

**R**. They can be written as the countable intersection of the open sets {*q*}*C*where*q*is rational.The set of rational numbers

**Q**is*not*a Gδ set in**R**. If**Q**were the intersection of open sets*An*, each*An*would be dense in**R**because**Q**is dense in**R**. However, the construction above gave the irrational numbers as a countable intersection of open dense subsets. Taking the intersection of both of these sets gives the empty set as a countable intersection of open dense sets in**R**, a violation of the Baire category theorem.The zero-set of a derivative of an everywhere differentiable real-valued function on

**R**is a Gδ set; it can be a dense set with empty interior, as shown by Pompeiu's construction.

A more elaborate example of a Gδ set is given by the following theorem:

**Theorem:**The setcontains a dense Gδsubset of the metric space. (SeeWeierstrass function#Density of nowhere-differentiable functions.)Properties

The notion of Gδ sets in metric (and topological) spaces is strongly related to the notion of completeness of the metric space as well as to the Baire category theorem. This is described by the Mazurkiewicz theorem:

**Theorem**(Mazurkiewicz): Letbe a complete metric space and. Then the following are equivalent:is a Gδ subset of

There is a metric on which is equivalent to such that is a complete metric space.

`A key property ofsets is that they are the possible sets at which a function from a topological space to a metric space iscontinuous. Formally: The set of points where a functionis continuous is aset. This is because continuity at a pointcan be defined by aformula, namely: For all positive integers, there is an open setcontainingsuch thatfor allin. If a value ofis fixed, the set offor which there is such a corresponding openis itself an open set (being a union of open sets), and theuniversal quantifieroncorresponds to the (countable) intersection of these sets. In the real line, the converse holds as well; for any Gδsubset`

*A*of the real line, there is a function*f*:**R**→**R**which is continuous exactly at the points in*A*. As a consequence, while it is possible for the irrationals to be the set of continuity points of a function (see thepopcorn function), it is impossible to construct a function which is continuous only on the rational numbers.`Inreal analysis, especiallymeasure theory,sets and their complements are also of great importance.`

Basic properties

The complement of a Gδ set is an Fσ set.

The intersection of countably many Gδ sets is a Gδ set, and the union of

*finitely*many Gδ sets is a Gδ set; a countable union of Gδ sets is called a Gδσ set.In metrizable spaces, every closed set is a Gδ set and, dually, every open set is an Fσ set.

A subspace

*A*of a completely metrizable space*X*is itself completely metrizable if and only if*A*is a Gδ set in*X*.A set that contains the intersection of a countable collection of dense open sets is called

**comeagre**or**residual.**These sets are used to define generic properties of topological spaces of functions.

The following results regard Polish spaces:^{[1]}

Let be a Polish topological space. Then a set is a Polish subspace (with respect to ) of if and only if it is a Gδ set.

Topological characterization of Polish spaces: If is a Polish space then it is homeomorphic to a Gδ subset of a compact metric space.

Gδ space

A **Gδ space** is a topological space in which every closed set is a Gδ set (Johnson 1970). A normal space which is also a Gδ space is **perfectly normal**. Every metrizable space is perfectly normal, and every perfectly normal space is completely normal: neither implication is reversible.

See also

Fσ set, the dual concept; note that "G" is German (

*Gebiet*) and "F" is French (*fermé*).*P*-space, any space having the property that every Gδ set is open

## References

*Measure Theory, Volume 4*. Petersburg, England: Digital Books Logistics. pp. 334–335. ISBN 0-9538129-4-4. Retrieved 1 April 2011.