Gδ set

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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 German with G for Gebiet (German: area, or neighbourhood) meaning open set in this case and δ for Durchschnitt (German: intersection).[citation needed] The term inner limiting set is also used. Gδ sets, and their dual, Fσ 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 Π0
2
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 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 continuity set of any real valued function is a Gδ subset of its domain (see the section properties for a more general and complete statement).
• 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:

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

Properties

The notion of Gδ 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.

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

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 finitely many Gδ sets is a Gδ set.
• A countable union of Gδ sets (which would be called a Gδσ set) is not a Gδ set in general. For example, the rational numbers Q do not form a Gδ set in R.
• In a topological space, the zero set of every real valued continuous function ${\displaystyle f}$  is a Gδ set, since ${\displaystyle f^{-1}(0)}$  is the intersection of the open sets ${\displaystyle \{x\in X:-1/n , ${\displaystyle (n=1,2,...)}$ .
• In a metrizable space, every closed set is a Gδ set and, dually, every open set is an Fσ set.[1] Indeed, a closed set ${\displaystyle F\subseteq X}$  is the zero set of the continuous function ${\displaystyle f(x)=d(x,F)}$ , where ${\displaystyle d}$  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 Gδ set.[2]
• A subspace A of a completely metrizable space X is itself completely metrizable if and only if A is a Gδ set in X.[3][4]

The following results regard Polish spaces:[5]

• Let ${\displaystyle X}$  be a Polish space. Then a subset ${\displaystyle G\subseteq X}$  with the subspace topology is Polish if and only if it is a Gδ set in ${\displaystyle X}$ .
• A topological space ${\displaystyle X}$  is Polish if and only if it is homeomorphic to a Gδ subset of a compact metric space.

Continuity set of real valued functions

A property of ${\displaystyle \mathrm {G_{\delta }} }$  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 such a function ${\displaystyle f}$  is continuous is a ${\displaystyle \mathrm {G_{\delta }} }$  set. This is because continuity at a point ${\displaystyle p}$  can be defined by a ${\displaystyle \Pi _{2}^{0}}$  formula, namely: For all positive integers ${\displaystyle n}$ , there is an open set ${\displaystyle U}$  containing ${\displaystyle p}$  such that ${\displaystyle d(f(x),f(y))<1/n}$  for all ${\displaystyle x,y}$  in ${\displaystyle U}$ . If a value of ${\displaystyle n}$  is fixed, the set of ${\displaystyle p}$  for which there is such a corresponding open ${\displaystyle U}$  is itself an open set (being a union of open sets), and the universal quantifier on ${\displaystyle n}$  corresponds to the (countable) intersection of these sets. In the real line, the converse holds as well; for any Gδ subset A of the real line, there is a function f: RR that 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 that is continuous only on the rational numbers.

Gδ space

A Gδ space[6] is a topological space in which every closed set is a Gδ set (Johnson 1970). A normal space that is also a Gδ space is called perfectly normal. For example, every metrizable space is perfectly normal.

• 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

Notes

1. ^ Willard, 15C, p. 105
2. ^ https://math.stackexchange.com/questions/1882733
3. ^ Willard, theorem 24.12, p. 179
4. ^ 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.
5. ^ Fremlin, p. 334
6. ^ 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.