In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Suslin representation of a set of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
s (more precisely, elements of
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
) is a
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
whose projection is that set of reals. More generally, a subset ''A'' of ''κ''
ω is ''λ''-Suslin if there is a
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
''T'' on ''κ'' × ''λ'' such that ''A'' = p
'T''
By a tree on ''κ'' × ''λ'' we mean here a subset ''T'' of the union of ''κ''
''i'' × ''λ''
''i'' for all ''i'' ∈ N (or ''i'' < ω in set-theoretical notation).
Here, p
'T''= is the projection of ''T'',
where
'T''= is the set of
branch
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk (botany), trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term '' ...
es through ''T''.
Since
'T''is a closed set for the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
on ''κ''
ω × ''λ''
ω where ''κ'' and ''λ'' are equipped with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
(and all closed sets in ''κ''
ω × ''λ''
ω come in this way from some tree on ''κ'' × ''λ''), ''λ''-Suslin subsets of ''κ''
ω are projections of closed subsets in ''κ''
ω × ''λ''
ω.
When one talks of ''Suslin sets'' without specifying the space, then one usually means Suslin subsets of R, which
descriptive set theorists usually take to be the set ω
ω.
See also
*
Suslin cardinal
*
Suslin operation In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers.
The Suslin operation was introduced by and . In Russia it is sometimes called the A-opera ...
External links
* R. Ketchersid
The strength of an ω1-dense ideal on ω1 under CH 2004.
Set theory
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