In mathematics, particularly

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, a G_δ space is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

in which

closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

s are in a way ‘separated’ from their complements using only countably many

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...

s. A G_δ space may thus be regarded as a space satisfying a different kind of

separation axiom In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometime ...

. In fact

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

G_δ spaces are referred to as

perfectly normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. Th ...

s, and satisfy the strongest of

separation axioms In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...

. G_δ spaces are also called perfect spaces. The term ''perfect'' is also used, incompatibly, to refer to a space with no

isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...

s; see

Perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all Limit point, limit points of S, also known as the derived se ...

Definition

A countable

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

of open sets in a topological space is called a G_δ set. Trivially, every open set is a G_δ set. Dually, a countable union of closed sets is called an F_σ set. Trivially, every closed set is an F_σ set. A topological space ''X'' is called a G_δ space if every closed subset of ''X'' is a G_δ set. Dually and equivalently, a ''G_δ space'' is a space in which every open set is an F_σ set.

Properties and examples

* Every subspace of a G_δ space is a G_δ space. * Every

metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric Metric or metrical may refer t ...

is a G_δ space. The same holds for

pseudometrizable space In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric ...

s. * Every

second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...

regular space is a G_δ space. This follows from the

Urysohn's metrization theorem In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...

in the Hausdorff case, but can easily be shown directly. * Every countable regular space is a G_δ space. * Every hereditarily Lindelöf regular space is a G_δ space. Such spaces are in fact

perfectly normal ''Perfectly Normal'' is a Canadian comedy film directed by Yves Simoneau, which premiered at the 1990 Festival of Festivals, before going into general theatrical release in 1991. Simoneau's first English-language film, it was written by Eugene Lip ...

. This generalizes the previous two items about second countable and countable regular spaces. * A G_δ space need not be normal, as R endowed with the

K-topology In mathematics, particularly topology, the K-topology is a Topological space, topology that one can impose on the set of all real numbers which has some interesting properties. Relative to the set of all real numbers carrying the standard topology, ...

shows. That example is not a regular space. Examples of Tychonoff G_δ spaces that are not normal are the

Sorgenfrey plane In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line \mathbb under the half-open interva ...

and the

Niemytzki plane In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. I ...

. * In a

first countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...

T₁ space, every

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance o ...

is a G_δ set. That is not enough for the space to be a G_δ space, as shown for example by the

lexicographic order topology on the unit square In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square ''S'', i.e. on the set of points (''x'',''y'') in the plane such that and Construction The ...

. * The

Sorgenfrey line In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...

is an example of a perfectly normal (i.e. normal G_δ) space that is not metrizable. * The

topological sum In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a Indexed family, family of topological spaces is a space formed by equipping the disjoi ...

X=_i X_i

of a family of disjoint topological spaces is a G_δ space if and only if each

X_i

is a G_δ space.

Notes

References

* * * Roy A. Johnson (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". ''The American Mathematical Monthly'', Vol. 77, No. 2, pp. 172–176
on JStor
{{DEFAULTSORT:Gdelta Space General topology Properties of topological spaces Real analysis