Gδ set
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 and .
Historically Gδ sets were also called inner limiting sets, but that terminology is not in use anymore.
Gδ sets, and their dual, F 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 Π sets of the Borel hierarchy.Examples
- Any open set is trivially a Gδ set.
- The irrational numbers are a Gδ set in the real numbers. They can be written as the countable intersection of the open sets where is rational.
- The set of rational numbers is a Gδ set in. If were the intersection of open sets each would be dense in because is dense in. 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, a violation of the Baire category theorem.
- The continuity set of any real valued function is a Gδ subset of its domain.
- The zero-set of a derivative of an everywhere differentiable real-valued function on is a Gδ set; it can be a dense set with empty interior, as shown by Pompeiu's construction.
- The set of functions in not differentiable at any point within contains a dense Gδ subset of the metric space.
Properties
Basic properties
- The complement of a Gδ set is an Fσ set, and vice versa.
- The intersection of countably many Gδ sets is a Gδ set.
- The union of many Gδ sets is a Gδ set.
- A countable union of Gδ sets is not a Gδ set in general. For example, the rational numbers do not form a Gδ set in.
- In a topological space, the zero set of every real valued continuous function is a Gδ set, since is the intersection of the open sets,.
- In a metrizable space, every closed set is a Gδ set and, dually, every open set is an Fσ set. Indeed, a closed set is the zero set of the continuous function, where indicates the distance from a point to a set. The same holds in pseudometrizable spaces.
- In a first countable T1 space, every singleton is a Gδ set.
- A subspace of a completely metrizable space is itself completely metrizable if and only if it is a Gδ set in.
- A subspace of a Polish space is itself Polish if and only if it is a Gδ set in. This follows from the previous result about completely metrizable subspaces and the fact that every subspace of a separable metric space is separable.
- A topological space is Polish if and only if it is homeomorphic to a Gδ subset of a compact metric space.
Continuity set of real valued functions
In the real line, the converse holds as well; for any Gδ subset of the real line, there is a function that is continuous exactly at the points in.