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 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. 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 ${\mathbf {\Pi }}_{2}^{0}$ sets of the Borel hierarchy.

Examples

Any open set is trivially a G_δ set
The irrational numbers are a G_δ set in the real numbers R. They can be written as the countable intersection of the sets {q}^C where q is rational.
The set of rational numbers Q is not a G_δ set in R. If Q were the intersection of open sets A_n, each A_n 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 G_δ set; it can be a dense set with empty interior, as shown by Pompeiu's construction.

A more elaborate example of a G_δ set is given by the following theorem:

Theorem: The set $D=\left\{f\in C([0,1]):f{\text{ is not differentiable at any point of }}[0,1]\right\}$ contains a dense G_δ subset of the metric space $C([0,1])$ . (See Weierstrass function#Density of nowhere-differentiable functions.)

Properties

The notion of G_δ 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:

Theorem (Mazurkiewicz): Let $({\mathcal {X}},\rho )$ be a complete metric space and $A\subset {\mathcal {X}}$ . Then the following are equivalent:

$A$ is a G_δ subset of ${\mathcal {X}}$
There is a metric $\sigma$ on $A$ which is equivalent to $\rho |A$ such that $(A,\sigma )$ is a complete metric space.

A key property of $G_\delta$ sets is that they are the possible sets at which a function from a topological space to a metric space is continuous. Formally: The set of points where a function $f$ is continuous is a $G_\delta$ set. This is because continuity at a point $p$ can be defined by a $\Pi _{2}^{0}$ formula, namely: For all positive integers $n$ , there is an open set $U$ containing $p$ such that $d(f(x),f(y))<1/n$ 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. In the real line, the converse holds as well; for any G_δ subset A of the real line, there is a function f: R → R which is continuous exactly at the points in A. 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.

in real analysis, especially measure theory, $G_\delta$ sets and their complements are also of great importance.

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 completely metrizable space X is itself completely metrizable 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.

The following results regard Polish spaces:^[1]

Let $({\mathcal {X}},{\mathcal {T}})$ be a Polish topological space and let $G\subset {\mathcal {X}}$ be a G_δ set (with respect to ${\mathcal {T}}$ ). Then $G$ is a Polish space with respect to the subspace topology on it.
Topological characterization of Polish spaces: If ${\mathcal {X}}$ is a Polish space then it is homeomorphic to a G_δ subset of a compact metric space.

G_δ space

A G_δ space is a topological space in which every closed set is a G_δ set (Johnson 1970). 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.

Notes

↑ Fremlin, D.H. (2003). "4, General Topology". Measure Theory, Volume 4. Petersburg, England: Digital Books Logistics. pp. 334–335. ISBN 0-9538129-4-4. Retrieved 1 April 2011.

References

Kelley, John L. (1955). General topology. van Nostrand. p. 134.
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446 P. 162.
Fremlin, D.H. (2003) [2003]. "4, General Topology". Measure Theory, Volume 4. Petersburg, England: Digital Books Logostics. ISBN 0-9538129-4-4. Retrieved 1 April 2011 P. 334.
Johnson, Roy A. (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". The American Mathematical Monthly. 77 (2): 172–176. doi:10.2307/2317335. JSTOR 2317335.

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[Fremlin_2003-1] Fremlin, D.H. (2003). "4, General Topology". Measure Theory, Volume 4. Petersburg, England: Digital Books Logistics. pp. 334–335. ISBN 0-9538129-4-4. Retrieved 1 April 2011.