G_δ 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 German with G for Gebiet (German: area, or neighbourhood) meaning open set in this case and $δ$ for Durchschnitt (German: intersection).^[1] Historically G_δ sets were also called inner limiting sets,^[2] but that terminology is not in use anymore. 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 Π⁰
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 the real numbers $\mathbb {R}$ . They can be written as the countable intersection of the open sets $\{q\}^{c}$ (the superscript denoting the complement) where $q$ is rational.
The set of rational numbers $\mathbb {Q}$ is not a G_δ set in $\mathbb {R}$ . If $\mathbb {Q}$ were the intersection of open sets $A_{n}$ each $A_{n}$ would be dense in $\mathbb {R}$ because $\mathbb {Q}$ is dense in $\mathbb {R}$ . 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 $\mathbb {R}$ , a violation of the Baire category theorem.
The continuity set of any real valued function is a G_δ subset of its domain (see the "Properties" section for a more general statement).
The zero-set of a derivative of an everywhere differentiable real-valued function on $\mathbb {R}$ is a G_δ set; it can be a dense set with empty interior, as shown by Pompeiu's construction.
The set of functions in $C([0,1])$ not differentiable at any point within $[0, 1]$ 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 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. $\mathrm {G_{\delta }}$ sets and their complements are also of importance in real analysis, especially measure theory.

Basic properties

The complement of a G_δ set is an F_σ set, and vice versa.
The intersection of countably many G_δ sets is a G_δ set.
The union of finitely many G_δ sets is a G_δ set.
A countable union of G_δ sets (which would be called a G_δσ set) is not a G_δ set in general. For example, the rational numbers $\mathbb {Q}$ do not form a G_δ set in $\mathbb {R}$ .
In a topological space, the zero set of every real valued continuous function $f$ is a (closed) G_δ set, since $f^{-1}(0)$ is the intersection of the open sets $\{x\in X:-1/n<f(x)<1/n\}$ , $(n=1,2,\ldots )$ .
In a metrizable space, every closed set is a G_δ set and, dually, every open set is an F_σ set.^[3] Indeed, a closed set $F\subseteq X$ is the zero set of the continuous function $f(x)=d(x,F)$ , where $d$ 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 G_δ set.^[4]
A subspace of a completely metrizable space $X$ is itself completely metrizable if and only if it is a G_δ set in $X$ .^[5]^[6]
A subspace of a Polish space $X$ is itself Polish if and only if it is a G_δ set in $X$ . 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 $X$ is Polish if and only if it is homeomorphic to a G_δ subset of a compact metric space.^[7]^[8]

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 $\mathrm {G_{\delta }}$ 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. 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 G_δ subset $A$ of the real line, there is a function $f:\mathbb {R} \to \mathbb {R}$ that is continuous exactly at the points in $A$ .^[9]

G_δ space

A G_δ space^[10] is a topological space in which every closed set is a G_δ set (Johnson 1970). A normal space that is also a G_δ space is called perfectly normal. For example, every metrizable space is perfectly normal.

Notes

^ Stein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, p. 23, ISBN 9781400835560.
^ Young, William; Young, Grace Chisholm (1906), Theory of Sets of Points, Cambridge University Press
^ Willard, 15C, p. 105
^ "General topology - when are singletons $G_\delta$?".
^ Willard, theorem 24.12, p. 179
^ Engelking, theorems 4.3.23 and 4.3.24 on p. 274. From the historical notes on p. 276, the forward implication was shown in a special case by S. Mazurkiewicz and in the general case by M. Lavrentieff; the reverse implication was shown in a special case by P. Alexandroff and in the general case by F. Hausdorff.
^ Fremlin, p. 334
^ The sufficiency of the condition uses the fact that every compact metric space is separable and complete, and hence Polish.
^ Saito, Shingo. "Properties of G_δ subsets of $\mathbb {R}$ " (PDF).
^ Steen & Seebach, p. 162

References

Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
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.
Fremlin, D.H. (2003) [2003]. "4, General Topology". Measure Theory, Volume 4. Petersburg, England: Digital Books Logostics. ISBN 0-9538129-4-4. Archived from the original on 1 November 2010. Retrieved 1 April 2011.
Willard, Stephen (2004) [1970], General Topology (Dover reprint of 1970 ed.), Addison-Wesley
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.

[ramtihs-1] Stein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, p. 23, ISBN 9781400835560.

[2] Young, William; Young, Grace Chisholm (1906), Theory of Sets of Points, Cambridge University Press

[3] Willard, 15C, p. 105

[4] "General topology - when are singletons $G_\delta$?".

[5] Willard, theorem 24.12, p. 179

[6] Engelking, theorems 4.3.23 and 4.3.24 on p. 274. From the historical notes on p. 276, the forward implication was shown in a special case by S. Mazurkiewicz and in the general case by M. Lavrentieff; the reverse implication was shown in a special case by P. Alexandroff and in the general case by F. Hausdorff.

[7] Fremlin, p. 334

[8] The sufficiency of the condition uses the fact that every compact metric space is separable and complete, and hence Polish.

[9] Saito, Shingo. "Properties of G_δ subsets of $\mathbb {R}$ " (PDF).

[10] Steen & Seebach, p. 162

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