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Gδ space

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In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘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.[1] The term perfect is also used, incompatibly, to refer to a space with no isolated points; see Perfect set.

Definition

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A countable intersection of open sets in a topological space is called a Gδ set. Trivially, every open set is a Gδ set. Dually, a countable union of closed sets is called an Fσ set. Trivially, every closed set is an Fσ set.

A topological space X is called a Gδ space[2] if every closed subset of X is a Gδ set. Dually and equivalently, a Gδ space is a space in which every open set is an Fσ set.

Properties and examples

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Notes

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  1. ^ Engelking, 1.5.H(a), p. 48
  2. ^ Steen & Seebach, p. 162
  3. ^ "General topology - Every regular and second countable space is a $G_\delta$ space, without assuming Urysohn's metrization theorem".
  4. ^ https://arxiv.org/pdf/math/0412558.pdf, lemma 6.1
  5. ^ "The Sorgenfrey plane is subnormal". 8 May 2014.
  6. ^ "General topology - Moore plane / Niemytzki plane and the closed $G_\delta$ subspaces".
  7. ^ "The Lexicographic Order and the Double Arrow Space". 8 October 2009.

References

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