#
G_{δ} set

In the mathematical field of topology, a **G _{δ} 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. G

_{δ}sets, and their dual, F

_{σ}sets, are the second level of the Borel hierarchy.

## Definition

In a topological space a **G _{δ} set** is a countable intersection of open sets. The G

_{δ}sets are exactly the level sets of the Borel hierarchy.

## Examples

- Any open set is trivially a G
_{δ}set. - The irrational numbers are a G
_{δ}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 G_{δ}set in**R**. If**Q**were the intersection of open sets*A*, each_{n}*A*would be dense in_{n}**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 continuity set of any real valued function is a G
_{δ}subset of its domain (see the section properties for a more general and complete statement). - The zero-set of a derivative of an everywhere differentiable real-valued function on
**R**is a G_{δ}set; it can be a dense set with empty interior, as shown by Pompeiu's construction.

A more elaborate example of a G_{δ} set is given by the following theorem:

**Theorem:** The set contains a dense G_{δ} subset of the metric space . (See Weierstrass function § Density of nowhere-differentiable functions.)

## Properties

The notion of G_{δ} 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): Let be a complete metric space and . Then the following are equivalent:

- is a G
_{δ}subset of - There is a metric on that is equivalent to such that is a complete metric space.

A key property of sets is that they are the possible sets at which a function from a topological space to a metric space is continuous. Formally: The set of points where a function is continuous is a set. This is because continuity at a point can be defined by a formula, namely: For all positive integers , there is an open set containing such that for all in . If a value of is fixed, the set of for which there is such a corresponding open is itself an open set (being a union of open sets), and the universal quantifier on corresponds to the (countable) intersection of these sets. In the real line, the converse holds as well; for any G_{δ} subset *A* of the real line, there is a function *f*: **R** → **R** that 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 the popcorn function), it is impossible to construct a function that is continuous only on the rational numbers.

In real analysis, especially measure theory, sets and their complements are also of great importance.

### Basic properties

- The complement of a G
_{δ}set is an F_{σ}set. - The intersection of countably many G
_{δ}sets is a G_{δ}set, and the union of*finitely*many G_{δ}sets is a G_{δ}set; a countable union of G_{δ}sets is called a G_{δσ}set. - In metrizable spaces, every closed set is a G
_{δ}set and, dually, every open set is an F_{σ}set. - A subspace
*A*of a completely metrizable space*X*is itself completely metrizable if and only if*A*is a G_{δ}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 G
_{δ}set. - Topological characterization of Polish spaces: If is a Polish space then it is homeomorphic to a G
_{δ}subset of a compact metric space.

## G_{δ} space

_{δ}space

A **G _{δ} space** is a topological space in which every closed set is a G

_{δ}set (Johnson 1970). A normal space that is also a G

_{δ}space is

**perfectly normal**. Every metrizable space is perfectly normal, and every perfectly normal space is completely normal: neither implication is reversible.

## See also

## Notes

- Fremlin, D.H. (2003). "4, General Topology".
*Measure Theory, Volume 4*. Petersburg, England: Digital Books Logistics. pp. 334–335. ISBN 0-9538129-4-4. Retrieved 1 April 2011.

## References

- Kelley, John L. (1955).
*General topology*. van Nostrand. p. 134. - Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978].
*Counterexamples in Topology*(Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446 P. 162. - Fremlin, D.H. (2003) [2003]. "4, General Topology".
*Measure Theory, Volume 4*. Petersburg, England: Digital Books Logostics. ISBN 0-9538129-4-4. Retrieved 1 April 2011 P. 334. - Johnson, Roy A. (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta".
*The American Mathematical Monthly*.**77**(2): 172–176. doi:10.2307/2317335. JSTOR 2317335.