- 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.
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.