In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Suslin representation of a set of
reals (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 ...
) 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, e.g., including only woody plants with secondary growth, only ...
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, e.g., including only woody plants with secondary growth, only ...
''T'' on ''κ'' × ''λ'' such that ''A'' = p
'T''
By a tree on ''κ'' × ''λ'' we mean a subset ''T'' ⊆ ⋃
''n''<ω(''κ''
''n'' × ''λ''
''n'') closed under initial segments, and p
'T''= is the projection of ''T'',
where
'T''= is the set of
branch
A branch, also called a ramus in botany, is a stem that grows off from another stem, or when structures like veins in leaves are divided into smaller veins.
History and etymology
In Old English, there are numerous words for branch, includ ...
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
In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2ω such that P is 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-operati ...
External links
* R. Ketchersid
The strength of an ω1-dense ideal on ω1 under CH 2004.
Set theory
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