# Mikhail Yakovlevich Suslin

Mikhail Suslin

Mikhail Yakovlevich Suslin (Russian: Михаи́л Я́ковлевич Су́слин; Krasavka, Saratov Oblast, November 15, 1894 – 21 October 1919, Krasavka) (sometimes transliterated Souslin) was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory.

His name is especially associated to Suslin's problem, a question relating to totally ordered sets that was eventually found to be independent of the standard system of set-theoretic axioms, ZFC.

He contributed greatly to the theory of analytic sets, sometimes called after him, a kind of a set of reals which is definable via trees. In fact, while he was a research student of Nikolai Luzin (in 1917) he found an error in an argument of Lebesgue, who believed he had proved that for any Borel set in ${\displaystyle \mathbb {R} ^{2}}$, the projection onto the real axis was also a Borel set.

Suslin died of typhus in the 1919 Moscow epidemic following the Russian Civil War.

## Publications

Suslin only published one paper during his life: a 4-page note.

• Souslin, M. Ya. (1917), "Sur une définition des ensembles mesurables B sans nombres transfinis", C. R. Acad. Sci. Paris, 164: 88–91
• Souslin, M. (1920), "Problème 3" (PDF), Fundamenta Mathematicae, 1: 223
• Souslin, M. Ya. (1923), Kuratowski, C., ed., "Sur un corps dénombrable de nombres réels", Fundamenta math. (in French), 4: 311–315, JFM 49.0147.03

1.  A Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition
2.  A Suslin cardinal is a cardinal λ such that there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ.
3.  The Suslin hypothesis says that Suslin lines do not exist
4.  A Suslin line is a complete dense unbounded totally ordered set satisfying the countable chain condition
5.  The Suslin number is the supremum of the cardinalities of families of disjoint open non-empty sets
6.  The Suslin operation, usually denoted by A, is an operation that constructs a set from a Suslin scheme
7.  The Suslin problem asks whether Suslin lines exist
8.  The Suslin property states that there is no uncountable family of pairwise disjoint non-empty open subsets
9.  A Suslin representation of a set of reals is a tree whose projection is that set of reals
10.  A Suslin scheme is a function with domain the finite sequences of positive integers
11.  A Suslin set is a set that is the image of a tree under a certain projection
12.  A Suslin space is the image of a Polish space under a continuous mapping
13.  A Suslin subset is a subset that is the image of a tree under a certain projection
14.  The Suslin theorem about analytic sets states that a set that is analytic and coanalytic is Borel
15.  A Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable.