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In mathematics, particularly topology, a G_δ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A G_δ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal G_δ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.

G_δ spaces are also called perfect spaces.^[1] The term perfect is also used, incompatibly, to refer to a space with no isolated points; see Perfect set.

Definition

A countable intersection of open sets in a topological space is called a G_δ set. Trivially, every open set is a G_δ set. Dually, a countable union of closed sets is called an F_σ set. Trivially, every closed set is an F_σ set.

A topological space X is called a G_δ space^[2] if every closed subset of X is a G_δ set. Dually and equivalently, a G_δ space is a space in which every open set is an F_σ set.

Properties and examples

Every subspace of a G_δ space is a G_δ space.
Every metrizable space is a G_δ space. The same holds for pseudometrizable spaces.
Every second countable regular space is a G_δ space. This follows from the Urysohn's metrization theorem in the Hausdorff case, but can easily be shown directly.^[3]
Every countable regular space is a G_δ space.
Every hereditarily Lindelöf regular space is a G_δ space.^[4] Such spaces are in fact perfectly normal. This generalizes the previous two items about second countable and countable regular spaces.
A G_δ space need not be normal, as R endowed with the K-topology shows. That example is not a regular space. Examples of Tychonoff G_δ spaces that are not normal are the Sorgenfrey plane^[5] and the Niemytzki plane.^[6]
In a first countable T₁ space, every singleton is a G_δ set. That is not enough for the space to be a G_δ space, as shown for example by the lexicographic order topology on the unit square.^[7]
The Sorgenfrey line is an example of a perfectly normal (i.e. normal G_δ) space that is not metrizable.
The topological sum $X={\coprod }_{i}X_{i}$ of a family of disjoint topological spaces is a G_δ space if and only if each $X_{i}$ is a G_δ space.

Notes

^ Engelking, 1.5.H(a), p. 48
^ Steen & Seebach, p. 162
^ "General topology - Every regular and second countable space is a $G_\delta$ space, without assuming Urysohn's metrization theorem".
^ https://arxiv.org/pdf/math/0412558.pdf, lemma 6.1
^ "The Sorgenfrey plane is subnormal". 8 May 2014.
^ "General topology - Moore plane / Niemytzki plane and the closed $G_\delta$ subspaces".
^ "The Lexicographic Order and the Double Arrow Space". 8 October 2009.

References

Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover Publications reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
Roy A. Johnson (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". The American Mathematical Monthly, Vol. 77, No. 2, pp. 172–176. on JStor

[1] Engelking, 1.5.H(a), p. 48

[2] Steen & Seebach, p. 162

[3] "General topology - Every regular and second countable space is a $G_\delta$ space, without assuming Urysohn's metrization theorem".

[4] ttps://arxiv.org/pdf/math/0412558.pdf, lemma 6.1

[5] "The Sorgenfrey plane is subnormal". 8 May 2014.

[6] "General topology - Moore plane / Niemytzki plane and the closed $G_\delta$ subspaces".

[7] "The Lexicographic Order and the Double Arrow Space". 8 October 2009.

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+{{Short description|Property of topological space}}
 {{DISPLAYTITLE:G<sub>δ</sub> space}}
-In [[mathematics]], particularly [[topology]], a '''G<sub>δ</sub> space''' a space in which [[closed set]]s are ‘separated’ from their complements using only countably many [[open set]]s. A G<sub>δ</sub> space may thus be regarded as a space satisfying a different kind of [[separation axiom]]. In fact [[normal space|normal]] G<sub>δ</sub> spaces are referred to as [[perfectly normal space]]s, and satisfy the strongest of [[separation axioms]].
+In mathematics, particularly [[topology]], a '''G<sub>δ</sub> space''' is a [[topological space]] in which [[closed set]]s are in a way ‘separated’ from their complements using only countably many [[open set]]s. A G<sub>δ</sub> space may thus be regarded as a space satisfying a different kind of [[separation axiom]]. In fact [[normal space|normal]] G<sub>δ</sub> spaces are referred to as [[perfectly normal space]]s, and satisfy the strongest of [[separation axioms]].
-G<sub>δ</sub> spaces are also called '''perfect spaces'''. The term ''perfect'' is also used, incompatibly, to refer to a space with no [[isolated point]]s; see [[perfect space]].
+G<sub>δ</sub> spaces are also called '''perfect spaces'''.<ref>Engelking, 1.5.H(a), p. 48</ref> The term ''perfect'' is also used, incompatibly, to refer to a space with no [[isolated point]]s; see [[Perfect set]].
 ==Definition==
-A subset of a [[topological space]] is said to be a [[G-delta set|G<sub>δ</sub> set]] if it can be written as the countable intersection of open sets. Trivially, any open subset of a topological space is a G<sub>δ</sub> set.
+A countable [[intersection]] of open sets in a topological space is called a [[G-delta set|G<sub>δ</sub> set]].  Trivially, every open set is a G<sub>δ</sub> set.  Dually, a countable union of closed sets is called an [[F-sigma set|F<sub>σ</sub> set]].  Trivially, every closed set is an F<sub>σ</sub> set.
-A topological space ''X'' is said to be a '''G<sub>δ</sub> space''' if every closed subspace of ''X'' is a G<sub>δ</sub> set (Steen and Seebach 1978, p.&nbsp;162).
+A topological space ''X'' is called a '''G<sub>δ</sub> space'''<ref>Steen & Seebach, p. 162</ref> if every closed subset of ''X'' is a G<sub>δ</sub> set. Dually and equivalently, a ''G<sub>δ</sub> space'' is a space in which every open set is an F<sub>σ</sub> set.
 ==Properties and examples==
-* In G<sub>δ</sub> spaces, every open set is the countable union of closed sets. In fact, a topological space is a G<sub>δ</sub>  space if and only if every open set is an [[F-sigma set|F<sub>&sigma;</sub> set]]
+* Every subspace of a G<sub>δ</sub> space is a G<sub>δ</sub> space.
+* Every [[metrizable space]] is a G<sub>δ</sub> space. The same holds for [[pseudometrizable space]]s.
+* Every [[second countable]] [[regular space|regular]] space is a G<sub>δ</sub> space.  This follows from the [[Urysohn's metrization theorem]] in the Hausdorff case, but can easily be shown directly.<ref>{{Cite web|url=https://math.stackexchange.com/q/1966730|title=General topology - Every regular and second countable space is a $G_\delta$ space, without assuming Urysohn's metrization theorem}}</ref>
+* Every countable regular space is a G<sub>δ</sub> space.
+* Every [[hereditarily Lindelöf]] regular space is a G<sub>δ</sub> space.<ref>https://arxiv.org/pdf/math/0412558.pdf, lemma 6.1</ref>  Such spaces are in fact [[normal space|perfectly normal]].  This generalizes the previous two items about second countable and countable regular spaces.
+* A G<sub>δ</sub> space need not be normal, as '''R''' endowed with the [[K-topology]] shows.  That example is not a regular space.  Examples of [[Tychonoff space|Tychonoff]] G<sub>δ</sub> spaces that are not normal are the [[Sorgenfrey plane]]<ref>{{Cite web|url=https://dantopology.wordpress.com/2014/05/07/the-sorgenfrey-plane-is-subnormal/|title = The Sorgenfrey plane is subnormal|date = 8 May 2014}}</ref> and the [[Niemytzki plane]].<ref>{{Cite web|url=https://math.stackexchange.com/q/2711463|title=General topology - Moore plane / Niemytzki plane and the closed $G_\delta$ subspaces}}</ref>
+* In a [[first countable]] [[T1 space|T<sub>1</sub> space]], every [[singleton (mathematics)|singleton]] is a G<sub>δ</sub>  set.  That is not enough for the space to be a G<sub>δ</sub> space, as shown for example by the [[lexicographic order topology on the unit square]].<ref>{{Cite web|url=https://dantopology.wordpress.com/2009/10/07/the-lexicographic-order-and-the-double-arrow-space/|title=The Lexicographic Order and the Double Arrow Space|date=8 October 2009}}</ref>
+* The [[Sorgenfrey line]] is an example of a perfectly normal (i.e. normal G<sub>δ</sub>) space that is not metrizable.
+* The [[topological sum]] <math>X={\coprod}_i X_i</math> of a family of disjoint topological spaces is a G<sub>δ</sub> space if and only if each <math>X_i</math> is a G<sub>δ</sub> space.
+==Notes==
-* Any [[metric space]] is a G<sub>δ</sub> space.
+ {{reflist}}
-* Without assuming Urysohn’s metrization theorem, one can prove that every [[regular space]] with a [[Second countable space|countable base]] is a G<sub>δ</sub> space.
-* A G<sub>δ</sub> space need not be normal, as '''R''' endowed with the [[K-topology]] shows.
-* In a [[First countable space|first countable]] T<sub>1</sub> space, any one point set is a G<sub>δ</sub>  set.
-* The [[Sorgenfrey line]] is an example of a perfectly normal (i.e normal G<sub>δ</sub> space) that is not metrizable
 ==References==
-* {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=507446}} | year=1995}}  P. 162.
+* {{cite book|last=Engelking|first=Ryszard| authorlink=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}}
+* {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover Publications reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}}
+* Roy A. Johnson (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". ''The American Mathematical Monthly'', Vol. 77, No. 2, pp.&nbsp;172–176. [https://www.jstor.org/stable/2317335 on JStor]
-* Roy A. Johnson (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". ''The American Mathematical Monthly'', Vol. 77, No. 2, pp.&nbsp;172-176. [http://www.jstor.org/stable/2317335 on JStor]
-{{DEFAULTSORT:Gδ Space}}
+{{DEFAULTSORT:Gdelta Space}}
 [[Category:General topology]]
+[[Category:Properties of topological spaces]]
 [[Category:Real analysis]]