G-delta set

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In general topology, a Gδ set is a subset of a topological space which is a countable intersection of open sets. An Fσ space is similarly a countable union of closed sets.

Properties

  • The pre-image of a Gδ set under a continuous map is again a Gδ set. In particular, the zero set of a continuous real-valued function is a Gδ set.
  • A closed Gδ set is a normal space is the zero set of a continuous real-valued function.
  • A Gδ in a complete metric space is again a complete metric space.

Gδ space

A Gδ space is a topological space in which every closed set is a Gδ set. A normal space which is also a Gδ space is perfectly normal. Every metrizable space is perfectly normal, and every perfectly normal space is a completely normal space; neither implication is reversible.

References