View source for Gδ space

{{Short description|Property of topological space}}
{{DISPLAYTITLE:G<sub>δ</sub> space}}
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'''.<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 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 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==
* 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==
 {{reflist}}

==References==
* {{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]

{{DEFAULTSORT:Gdelta Space}}
[[Category:General topology]]
[[Category:Properties of topological spaces]]
[[Category:Real analysis]]