# G-delta set

From Citizendium

In general topology, a **G _{δ} set** is a subset of a topological space which is a countable intersection of open sets. An

**F**space is similarly a countable union of closed sets.

_{σ}## 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.