Gδ space

In mathematics, particularly topology, a G-delta space a space in which closed sets are ‘separated’ from their complements using only countably many open sets. A G-delta space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal G-delta spaces are referred to as perfectly normal spaces and such spaces satisfy the strongest of separation axioms. The formal definition will be given after some terminology.

Terminology

• If A is a subset of a topological space, then A is said to be a G-delta set if A can be written as the countable intersection of open sets. Trivially, any open subset of a topological space is a G-delta set.

Formal definition

If X is a topological space, then X is said to be a G-delta space if every closed subspace of X is a G-delta set.

Properties and Examples

• In G-delta spaces, every open set is the countable union of closed sets. In fact, a topological space is a G-delta space iff every open set is an F-sigma set

• Any metric space is a G-delta space.

• Without assuming Urysohn’s metrization theorem, one can prove that every regular space with a countable base is a G-delta space.

• A G-delta 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-delta set.

• The Sorgenfrey line is an example of a perfectly normal (i.e normal G-delta space) that is not metrizable

See also

• Separation axioms
• G-delta set
• F-sigma set
• Normal space
• Metric space

References

A compact non-metrizable space such that every closed set is a G-delta set