Gδ set

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 ' ("German": area) meaning open set in this case and δ for ' ("German": intersection). The term inner limiting set is also used. G_δ sets, and their dual F_σ sets, are the second level of the Borel hierarchy.

Definition

In a topological space a G_δ set is a countable intersection of open sets. The G_δ sets are exactly the level $mathbf\{Pi\}^0\_2$ sets of the Borel hierarchy.

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.

Properties

A key property of G_δ sets is that they are the possible sets at which a function between metric spaces is continuous. Formally:

The set of points where a function "f" is continuous is a G_δ set.

This is because continuity at a point "p" can be defined by a $Pi^0\_2$ formula, namely .The formula states that for every natural number $epsilon\; >\; 0$, there exists a natural number $N\; >\; 0$ such that whenever , we have . If you fix a value of $epsilon$, the set of "x" for which there is a corresponding "N" is an open set, and the universal quantifier on the $epsilon$ corresponds to the 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 which is continuous only on the rational numbers.

Basic 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; a countable union of G_δ sets is called a G_δσ 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 or residual. These sets are used to define generic properties of topological spaces of functions.

G_δ space

A G_δ space is a topological space in which every closed set is a G_δ set.Fact|date=August 2008 A normal space which is also a G_δ space is perfectly normal. Every metrizable space is perfectly normal, and every perfectly normal space is completely normal: neither implication is reversible.

See also

* F_σ set, the dual concept; note that "G" is German (') and "F" is French (').

References

* John L. Kelley, "General topology", van Nostrand, 1955. P.134.
* | year=1995 P. 162.