Gδ 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) 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 R, the real numbers, as they can be written as the intersection over all rational numbers q of the complement of {q} in R.

• The rational numbers Q are not a G_δ set. If we were able to write Q as the intersection of open sets A_n, each A_n would have to be dense in R since 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.

Properties

A key property of G_δ 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 f is continuous is a G_δ set.

This is because continuity at a point p can be defined by a formula, namely:

For all positive integer n, there is an open set U containing p such that d(f(x),f(y)) < 1 / n for all x,y in U.

If a value of n is fixed, the set of p for which there is such a corresponding open U is itself an open set (being a union of open sets), and the universal quantifier on n 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 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 the popcorn function), it is impossible to construct a function which is continuous only on the rational numbers.

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 topologically complete space X is itself topologically complete 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.

G_δ space

A G_δ space is a topological space in which every closed set is a G_δ set.^{[citation needed]} A normal space which 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

• F_σ set, the dual concept; note that "G" is German (Gebiet) and "F" is French (fermé).

References

• John L. Kelley, General topology, van Nostrand, 1955. P.134.
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, MR507446, ISBN 978-0-486-68735-3  P. 162.