G_δ space

In mathematics, particularly topology, a G_δ space is a space in which closed sets are ‘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. The term perfect is also used, incompatibly, to refer to a space with no isolated points; see Perfect space.

Definition

A subset of a topological space is said to be a G_δ set if it can be written as the countable intersection of open sets. Trivially, any open subset of a topological space is a G_δ set.

A topological space X is said to be a G_δ space if every closed subspace of X is a G_δ set (Steen and Seebach 1978, p. 162).

Properties and examples

In G_δ spaces, every open set is the countable union of closed sets. In fact, a topological space is a G_δ space if and only if every open set is an F_σ set.
Any metric space is a G_δ space.
Without assuming Urysohn's metrization theorem, one can prove that every regular space with a countable base is a G_δ space.
A G_δ space need not be normal, as R endowed with the K-topology shows.
In a first-countable T₁ space, any one-point set is a G_δ set.
The Sorgenfrey line is an example of a perfectly normal (i.e. normal G_δ) space that is not metrizable.

References

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 P. 162.
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