Gδ 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.