G-delta space

In mathematics, particularly topology, a G_δ space 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

* | year=1995 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. [http://www.jstor.org/stable/2317335 on JStor]