G-delta space
Encyclopedia
In mathematics
, particularly topology
, a Gδ space a space in which closed set
s are ‘separated’ from their complements using only countably many open set
s. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom
. In fact normal
Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.
Gδ spaces are also called perfect spaces. The term perfect is also used, incompatibly, to refer to a space with no isolated point
s; see perfect space
.
is said to be a Gδ set
if it can be written as the countable intersection of open sets. Trivially, any open subset of a topological space is a Gδ set.
A topological space X is said to be a Gδ space if every closed subspace of X is a Gδ set (Steen and Seebach 1978, p. 162).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a Gδ space a space in which closed set
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...
s are ‘separated’ from their complements using only countably many open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom
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...
. In fact normal
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...
Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.
Gδ spaces are also called perfect spaces. The term perfect is also used, incompatibly, to refer to a space with no isolated point
Isolated point
In topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S.In particular, in a Euclidean space ,...
s; see perfect space
Perfect space
In mathematics, in the field of topology, perfect spaces are spaces that have no isolated points. In such spaces, every point can be approximated arbitrarily well by other points - given any point and any topological neighborhood of the point, there is another point within the neighborhood.The...
.
Definition
A subset of a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is said to be a Gδ set
G-delta set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
if it can be written as the countable intersection of open sets. Trivially, any open subset of a topological space is a Gδ set.
A topological space X is said to be a Gδ space if every closed subspace of X is a Gδ set (Steen and Seebach 1978, p. 162).
Properties and examples
- In Gδ spaces, every open set is the countable union of closed sets. In fact, a topological space is a Gδ space if and only if every open set is an Fσ setF-sigma setIn mathematics, an Fσ set is a countable union of closed sets. The notation originated in France with F for fermé and σ for somme ....
- Any metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
is a Gδ space.
- Without assuming Urysohn’s metrization theorem, one can prove that every regular spaceRegular spaceIn topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
with a countable base is a Gδ space.
- A Gδ space need not be normal, as R endowed with the K-topology shows.
- In a first countable T1 space, any one point set is a Gδ set.
- The Sorgenfrey line is an example of a perfectly normal (i.e normal Gδ space) that is not metrizable