Gδ set
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 in Germany with G for Gebiet (German: area, or neighbourhood) meaning open set in this case and δ for Durchschnitt (German: intersection). The term inner limiting set is also used. Gδ sets, and their dual Fσ sets, are the second level of the Borel hierarchy.
Definition
Examples
Any open set is trivially a Gδ set
The irrational numbers are a Gδ set in the real numbers R. They can be written as the countable intersection of the open sets {q}C where q is rational.
The set of rational numbers Q is not a Gδ set in R. If Q were the intersection of open sets An, each An would be dense in R because 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 zero-set of a derivative of an everywhere differentiable real-valued function on R is a Gδ set; it can be a dense set with empty interior, as shown by Pompeiu's construction.
A more elaborate example of a Gδ set is given by the following theorem:
![D=\left\{f\in C([0,1]):f{\text{ is not differentiable at any point of }}[0,1]\right\}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60902342bbd3575f2b2db49b41e297a3f9bbb8b3)
![C([0,1])](https://wikimedia.org/api/rest_v1/media/math/render/svg/44211c4c325ea7edb9462e7ccecda09841a41216)
Properties
The notion of Gδ sets in metric (and topological) spaces is strongly related to the notion of completeness of the metric space as well as to the Baire category theorem. This is described by the Mazurkiewicz theorem:
is a Gδ subset of
There is a metric on which is equivalent to such that is a complete metric space.
Basic properties
The complement of a Gδ set is an Fσ set.
The intersection of countably many Gδ sets is a Gδ set, and the union of finitely many Gδ sets is a Gδ set; a countable union of Gδ sets is called a Gδσ set.
In metrizable spaces, every closed set is a Gδ set and, dually, every open set is an Fσ set.
A subspace A of a completely metrizable space X is itself completely metrizable if and only if A is a Gδ 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.
The following results regard Polish spaces:[1]
Let be a Polish topological space. Then a set is a Polish subspace (with respect to ) of if and only if it is a Gδ set.
Topological characterization of Polish spaces: If is a Polish space then it is homeomorphic to a Gδ subset of a compact metric space.
Gδ space
A Gδ space is a topological space in which every closed set is a Gδ set (Johnson 1970). A normal space which is also a Gδ space is perfectly normal. Every metrizable space is perfectly normal, and every perfectly normal space is completely normal: neither implication is reversible.
See also
Fσ set, the dual concept; note that "G" is German (Gebiet) and "F" is French (fermé).
P-space, any space having the property that every Gδ set is open