G-delta set

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

Properties

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

Gδ space

A G_δ space is a topological space in which every closed set is a G_δ set. A normal space which is also a G_δ 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

• J.L. Kelley (1955). General topology. van Nostrand, 134,207-208. 
• Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 162. ISBN 0-387-90312-7.