Gδ space

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In mathematics, particularly topology, a Gδ space 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[edit]

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[edit]

  • 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
  • A Gδ space need not be normal, as R endowed with the K-topology shows.
  • The Sorgenfrey line is an example of a perfectly normal (i.e. normal Gδ space) that is not metrizable

References[edit]

  • 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