Gδ space

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.

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

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.
gollark: Ready to launch heavlisp into the nearest stellar core.
gollark: Well, I came up with a quick hacky fix so we'll be ready to launch soon.
gollark: Oh right, the VFS thing.
gollark: What *is* heavlisp? Should I be worried?
gollark: Why does potatOS have a half-finished heavlisp integration?

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
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