G_δ set

• The complement of a G_δ set is an F_σ set, and vice versa.
• The intersection of countably many G_δ sets is a G_δ set.
• The union of finitely many G_δ sets is a G_δ set.
• A countable union of G_δ sets (which would be called a G_δσ set) is not a G_δ set in general. For example, the rational numbers Q do not form a G_δ set in R.
• In a topological space, the zero set of every real valued continuous function ${\displaystyle f}$ is a G_δ set, since ${\displaystyle f^{-1}(0)}$ is the intersection of the open sets ${\displaystyle \{x\in X:-1/n<f(x)<1/n\}}$, ${\displaystyle (n=1,2,...)}$.
• In a metrizable space, every closed set is a G_δ set and, dually, every open set is an F_σ set.[1] Indeed, a closed set ${\displaystyle F\subseteq X}$ is the zero set of the continuous function ${\displaystyle f(x)=d(x,F)}$, where ${\displaystyle d}$ indicates the distance from a point to a set. The same holds in pseudometrizable spaces.
• In a first countable T₁ space, every singleton is a G_δ set.[2]
• A subspace A of a completely metrizable space X is itself completely metrizable if and only if A is a G_δ set in X.[3][4]

The following results regard Polish spaces:[5]

• Let ${\displaystyle X}$ be a Polish space. Then a subset ${\displaystyle G\subseteq X}$ with the subspace topology is Polish if and only if it is a G_δ set in ${\displaystyle X}$.
• A topological space ${\displaystyle X}$ is Polish if and only if it is homeomorphic to a G_δ subset of a compact metric space.

A property of ${\displaystyle \mathrm {G_{\delta }} }$ sets is that they are the possible sets at which a function from a topological space to a metric space is continuous. Formally: The set of points where such a function ${\displaystyle f}$ is continuous is a ${\displaystyle \mathrm {G_{\delta }} }$ set. This is because continuity at a point ${\displaystyle p}$ can be defined by a ${\displaystyle \Pi _{2}^{0}}$ formula, namely: For all positive integers ${\displaystyle n}$, there is an open set ${\displaystyle U}$ containing ${\displaystyle p}$ such that ${\displaystyle d(f(x),f(y))<1/n}$ for all ${\displaystyle x,y}$ in ${\displaystyle U}$. If a value of ${\displaystyle n}$ is fixed, the set of ${\displaystyle p}$ for which there is such a corresponding open ${\displaystyle U}$ is itself an open set (being a union of open sets), and the universal quantifier on ${\displaystyle n}$ corresponds to the (countable) intersection of these sets. In the real line, the converse holds as well; for any G_δ subset A of the real line, there is a function f: R → R that is continuous exactly at the points in A. As a consequence, while it is possible for the irrationals to be the set of continuity points of a function (see the popcorn function), it is impossible to construct a function that is continuous only on the rational numbers.