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 ${\displaystyle X={\coprod }_{i}X_{i}}$ of a family of disjoint topological spaces is a G_δ space if and only if each ${\displaystyle X_{i}}$ is a G_δ space.

Notes

1. Engelking, 1.5.H(a), p. 48
2. Steen & Seebach, p. 162
3. https://math.stackexchange.com/questions/1966730
4. https://arxiv.org/pdf/math/0412558.pdf, lemma 6.1
5. https://dantopology.wordpress.com/2014/05/07/the-sorgenfrey-plane-is-subnormal/
6. https://math.stackexchange.com/questions/2711463
7. https://dantopology.wordpress.com/2009/10/07/the-lexicographic-order-and-the-double-arrow-space/

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