In the mathematical field of topology a G-delta set or G_δ set is a set in a topological space which is in a certain sense simple. The notation originated in Germany with G for Gebiet (german:area) meaning open set in this case and δ for Durchschnitt (german:intersection).

Definition

In a topological space a G_δ set is a countable intersection of open sets.

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. If we were able to write Q as the intersection of 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.

• The set of points at which a function from R to itself is continuous can be shown to be a G_δ.

Thus while it may be possible for the irrationals to be the set of continuity points of a function (in fact, such a function does exist), it is impossible to construct a function which is continuous only on the rational numbers.

Properties

• In metrizable spaces, every closed set is a G_δ set.

• The complement of a G_δ set is an F_σ. In a metrizable space, every open 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 subspace A of a topologically complete space X is itself topologically complete iff A is a G_δ set in X

See also

• F-sigma set, the dual concept