G_δ space

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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