(which would be called a Gδσ set) is not a Gδ set in general. For example, the rational numbers do not form a Gδ set in . In a topological space, the zero set of every real valued continuous function is a (closed) Gδ set, since is the intersection of the open sets , . In a metrizable space, every closed set is a Gδ set and, dually, every open set is an Fσ set. Indeed, a closed set is the zero set of the continuous function , where 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 Gδ set. A subspace of a completely metrizable space is itself completely metrizable if and only if it is a Gδ set in . A subspace of a Polish space is itself Polish if and only if it is a Gδ set in . This follows from the previous result about completely metrizable subspaces and the fact that every subspace of a separable metric space is separable. A topological space is Polish if and only if it is homeomorphic to a Gδ subset of a compact metric space. Continuity set of real valued functions The set of points where a function from a topological space to a metric space is continuous is a set. This is because continuity at a point can be defined by a formula, namely: For all positive integers there is an open set containing such that for all in . If a value of is fixed, the set of for which there is such a corresponding open is itself an open set (being a union of open sets), and the universal quantifier on corresponds to the (countable) intersection of these sets. 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. In the real line, the converse holds as well; for any Gδ subset of the real line, there is a function that is continuous exactly at the points in . Gδ space A Gδ space is a topological space in which every closed set is a Gδ set . A normal space that is also a Gδ space is called perfectly normal.