In the mathematical field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a G
δ set is a
subset of 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 po ...
that is a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
intersection of
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 su ...
s. The notation originated in
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
with ''G'' for ''
Gebiet'' (''German'': area, or neighbourhood) meaning open set in this case and for ''
Durchschnitt'' (''German'': intersection).
[.]
Historically G
δ sets were also called inner limiting sets, but that terminology is not in use anymore.
G
δ sets, and their dual,
F sets, are the second level of the
Borel hierarchy In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...
.
Definition
In a topological space a G
δ set is a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
intersection of
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 su ...
s. The G
δ sets are exactly the level Π sets of the
Borel hierarchy In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...
.
Examples
* Any open set is trivially a G
δ set.
* The
irrational numbers
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
are a G
δ set in the real numbers
. They can be written as the countable intersection of the open sets
(the superscript denoting the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
) where
is
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
.
* The set of rational numbers
is a G
δ set in
. If
were the intersection of open sets
each
would be
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in
because
is dense in
. However, the construction above gave the irrational numbers as a countable intersection of open dense subsets. Taking the intersection of both of these sets gives the
empty set as a countable intersection of open dense sets in
, a violation of the
Baire category theorem.
* The
continuity set In measure theory, a branch of mathematics, a continuity set of a measure ''μ'' is any Borel set ''B'' such that
: \mu(\partial B) = 0\,,
where \partial B is the (topological) boundary of ''B''. For signed measures, one asks that
: , \mu, (\ ...
of any real valued function is a G
δ subset of its domain (see the "Properties" section for a more general statement).
* The zero-set of a
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of an everywhere differentiable real-valued function on
is a G
δ set; it can be a dense set with empty interior, as shown by
Pompeiu's construction.
* The set of functions in
not differentiable at any point within contains a dense G
δ subset of the metric space
. (See .)
Properties
The notion of G
δ sets in
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
(and
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
) 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.
sets and their complements are also of importance in
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, especially
measure theory.
Basic properties
* The
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of a G
δ set is an
Fσ set, and vice versa.
* The intersection of countably many G
δ sets is a G
δ set.
* The union of 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
do not form a G
δ set in
.
* In a topological space, the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or e ...
of every real valued continuous function
is a (closed) G
δ set, since
is the intersection of the open sets
,
.
* In a
metrizable
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) s ...
space, every
closed set is a G
δ set and, dually, every open set is an F
σ 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
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 ...
is a G
δ set.
* A
subspace of a
completely metrizable In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' ind ...
space
is itself completely metrizable if and only if it is a G
δ set in
.
* A subspace of a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named be ...
is itself Polish if and only if it is a G
δ 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 G
δ subset of a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
.
Continuity set of real valued functions
The set of points where a function
from a topological space to a metric space is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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 G
δ subset
of the real line, there is a function
that is continuous exactly at the points in
.
Gδ space
A
Gδ space[Steen & Seebach, p. 162] is a topological space in which every
closed set is a G
δ set . A
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. T ...
that is also a G
δ space is called
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 ...
. For example, every metrizable space is perfectly normal.
See also
*
Fσ set, the
dual concept; note that "G" is German (''
Gebiet'') and "F" is French (''
fermé'').
*
''P''-space, any space having the property that every G
δ set is open
Notes
References
*
*
*
*
*
*
{{DEFAULTSORT:G Set
General topology
Descriptive set theory