In mathematics, particularly topology, a Gδ space is a space in which closed sets are ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.
A topological space X is said to be a Gδ space if every closed subspace of X is a Gδ set (Steen and Seebach 1978, p. 162).
Properties and examples
- In Gδ spaces, every open set is the countable union of closed sets. In fact, a topological space is a Gδ space if and only if every open set is an Fσ set
- Any metric space is a Gδ space.
- Without assuming Urysohn’s metrization theorem, one can prove that every regular space with a countable base is a Gδ space.
- A Gδ 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 Gδ set.
- The Sorgenfrey line is an example of a perfectly normal (i.e. normal Gδ space) that is not metrizable
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) , Counterexamples in Topology (Dover Publications reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 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