Gδ set

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The correct title of this article is G_δ set. It features superscript or subscript characters that are substituted or omitted because of technical limitations.

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). G_δ sets, and their dual F_σ sets, are the second level of the Borel hierarchy.

1 Definition
2 Examples
3 Properties
4 See also

[edit] Definition

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

[edit] 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.

The set of points at which a function from R to itself is continuous is a G_δ set. Thus 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.

[edit] 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.

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. These sets are used to define generic properties of topological spaces of functions.