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 from the German nouns open set and intersection. Historically Gδ sets were also called inner limiting sets, but that terminology is not in use anymore. Gδ sets, and their dual, F𝜎 sets, are the second level of the Borel hierarchy.

Definition

In a topological space a Gδ set is a countable intersection of open sets. The Gδ sets are exactly the level Π0 2 sets of the Borel hierarchy.

Examples

Properties

The notion of Gδ sets in metric (and topological) spaces is related to the notion of completeness of the metric space as well as to the Baire category theorem. See the result about completely metrizable spaces in the list of properties below. sets and their complements are also of importance in real analysis, especially measure theory.

Basic properties

Continuity set of real valued functions

The set of points where a function f from a topological space to a metric space is continuous is a set. This is because continuity at a point p can be defined by a \Pi^0_2 formula, namely: For all positive integers n, there is an open set U containing p such that 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. 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 the real line, the converse holds as well; for any Gδ subset A of the real line, there is a function that is continuous exactly at the points in A.

Gδ space

A Gδ space is a topological space in which every closed set is a Gδ set. A normal space that is also a Gδ space is called perfectly normal. For example, every metrizable space is perfectly normal.