Gδ set
In the mathematical field of topology, a G set is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with "G" for "Gebiet" ("German": area, or neighbourhood) meaning open set in this case and for "Durchschnitt" ("German": intersection). Historically G sets were also called inner limiting sets, but that terminology is not in use anymore. G sets, and their dual, F sets, are the second level of the Borel hierarchy. Definition. In a topological space a G set is a countable intersection of open sets. The G sets  are exactly the level Π sets of the Borel hierarchy. Properties. The notion of G sets in metric (and topological) spaces is related to the notion of completeness of the metric space as well as to the Baire category theorem. See the result about completely metrizable spaces in the list of properties below. formula_16 sets and their complements are also of importance in real analysis, especially measure theory. Continuity set of real valued functions. The set of points where a function formula_19 from a topological space to a metric space is continuous is a formula_16 set. This is because continuity  at a point formula_33 can be defined by a formula_34 formula, namely: For all positive integers formula_35 there is an open set formula_36 containing formula_33 such that formula_38 for all formula_39 in formula_36. If a value of formula_41 is fixed, the set of formula_33 for which there is such a corresponding open formula_36 is itself an open set (being a union of open sets), and the universal quantifier on formula_41 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 G subset formula_45 of the real line, there is a function formula_46 that is continuous exactly at the points in formula_45. G space. A G space is a topological space in which every closed set is a G set . A normal space that is also a G space is called perfectly normal. For example, every metrizable space is perfectly normal.