F-sigma set
Encyclopedia
In mathematics, an Fσ set (said F-sigma set) is a countable union
of closed set
s. The notation originated in France
with F for fermé (French
: closed) and σ for somme (French: sum, union).
In metrizable spaces, every closed set
is an Fσ set.
The complement of an Fσ set is a Gδ set
. In a metrizable space, any closed set is a Gδ set.
The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set.
The set
of rationals is an Fσ set. The set 
of irrationals is not a Fσ set.
In a Tychonoff space, each enumerable set is an Fσ set, because a point
is closed.
For example, the set
of all point
s
in the Cartesian plane such that 
is rational
is an Fσ set because it can be expressed as the union of all the line
s passing through the origin
with rational slope:
where
, is the set of rational numbers, which is a countable set.
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of 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. The notation originated in France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
with F for fermé (French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
: closed) and σ for somme (French: sum, union).
In metrizable spaces, every 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...
is an Fσ set.
The complement of an Fσ set is 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...
. In a metrizable space, any closed set is a Gδ set.
The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set.
Examples
Each closed set is an Fσ set.The set
of rationals is an Fσ set. The set of irrationals is not a Fσ set.In a Tychonoff space, each enumerable set is an Fσ set, because a point
is closed.For example, the set
of all pointPoint (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
s
in the Cartesian plane such that is rationalRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
is an Fσ set because it can be expressed as the union of all the line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
s passing through the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
with rational slope:
where
, is the set of rational numbers, which is a countable set.See also
- Gδ setG-delta setIn 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...
— the dualDuality (mathematics)In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
notion. - Borel hierarchyBorel hierarchyIn 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 the rank of the Borel set...