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Mikhail Yakovlevich Suslin
Mikhail Yakovlevich Suslin (russian: Михаи́л Я́ковлевич Су́слин; , 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. Biography Mikhail Suslin was born on November 15, 1894 in the village of Krasavka, the only child of poor peasants Yakov Gavrilovich and Matrena Vasil'evna Suslin. From a young age, Suslin showed a keen interest in mathematics and was encouraged to continue his education by his primary school teacher, Vera Andreevna Teplogorskaya-Smirnova. From 1905 to 1913 he attended Balashov boys' grammar school. In 1913, Suslin enrolled at the Imperial Moscow University and studied under the tutelage of Nikolai Luzin. He graduated with a degree in mathematics in 1917 and immediately began working at the Ivanovo-Voznesensk Polytechnic Institute. Suslin died of typhus in the 1919 Moscow epidemic follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saratov Oblast
Saratov Oblast (russian: Сара́товская о́бласть, ''Saratovskaya oblast'') is a federal subjects of Russia, federal subject of Russia (an oblast), located in the Volga Federal District. Its administrative center is the types of inhabited localities in Russia, city of Saratov. As of the Russian Census (2010), 2010 Census, its population was 2,521,892. Geography The oblast is located in the southeast of European Russia, in the northern part of the Lower Volga region. From west to east its territory stretches for , and from north to south for . The highest point of Saratov Oblast is an unnamed hill of the Khvalynsk Mountains reaching above sea level. The oblast borders on: * Volgograd Oblast to the south * Voronezh Oblast, Voronezh and Tambov Oblast, Tambov oblasts to the west * Penza Oblast, Penza, Samara Oblast, Samara and Ulyanovsk Oblast, Ulyanovsk oblasts to the north; * Kazakhstan (West Kazakhstan Region) to the east Natural resources Of particular ag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (descriptive Set Theory)
In descriptive set theory, a tree on a set X is a collection of finite sequences of elements of X such that every prefix of a sequence in the collection also belongs to the collection. Definitions Trees The collection of all finite sequences of elements of a set X is denoted X^. With this notation, a tree is a nonempty subset T of X^, such that if \langle x_0,x_1,\ldots,x_\rangle is a sequence of length n in T, and if 0\le m and called the ''body'' of the tree T. A tree that has no branches is called ''wellfounded''; a tree with at least one branch is ''illfounded''. By Kőnig's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded. Terminal nodes A finite sequence that belongs to a tree T is called a terminal node if it is not a prefix of a longer sequence in T. Equivalently, \langle x_0,x_1,\ldots,x_\rangle \in T is terminal if there is no element x of X such that that \langle x_0,x_1,\ldots,x_,x\rangle \in T. A tree that does not h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Set
In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset ''A'' of ''κ''ω is ''λ''-Suslin if there is a tree ''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 es through ''T''. Since 'T''is a closed set for the |
Suslin Scheme
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-operation after Alexandrov. It is usually denoted by the symbol 𝓐 (a calligraphic capital letter A). Definitions A Suslin scheme is a family P = \ of subsets of a set X indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set :\mathcal A P = \bigcup_ \bigcap_ P_ Alternatively, suppose we have a Suslin scheme, in other words a function M from finite sequences of positive integers n_1,\dots, n_k to sets M_. The result of the Suslin operation is the set : \mathcal A(M) = \bigcup \left(M_ \cap M_ \cap M_ \cap \dots \right) where the union is taken over all infinite sequences n_1,\dots, n_k, \dots If M is a family of subsets of a set X, then \mathcal A(M) is the family of subsets of X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Representation
In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset ''A'' of ''κ''ω is ''λ''-Suslin if there is a tree ''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 es through ''T''. Since 'T''is a closed set for the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Property
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set ''R'' with the four properties # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two distinct elements there is another); # the order on ''R'' is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and # every collection of mutually disjoint non-empty open intervals in ''R'' is countable (this is the countable chain condition for the order topology of ''R''), is ''R'' necessarily order-isomorphic to the real line R? ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Problem
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set ''R'' with the four properties # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two distinct elements there is another); # the order on ''R'' is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and # every collection of mutually disjoint non-empty open intervals in ''R'' is countable (this is the countable chain condition for the order topology of ''R''), is ''R'' necessarily order-isomorphic to the real line R? ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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-operation after Alexandrov. It is usually denoted by the symbol 𝓐 (a calligraphic capital letter A). Definitions A Suslin scheme is a family P = \ of subsets of a set X indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set :\mathcal A P = \bigcup_ \bigcap_ P_ Alternatively, suppose we have a Suslin scheme, in other words a function M from finite sequences of positive integers n_1,\dots, n_k to sets M_. The result of the Suslin operation is the set : \mathcal A(M) = \bigcup \left(M_ \cap M_ \cap M_ \cap \dots \right) where the union is taken over all infinite sequences n_1,\dots, n_k, \dots If M is a family of subsets of a set X, then \mathcal A(M) is the family of subsets of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Number
In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. Cardinal functions in set theory * The most frequently used cardinal function is a function that assigns to a set ''A'' its cardinality, denoted by , ''A'' , . * Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers. * Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers. * Cardinal characteristics of a (proper) ideal ''I'' of subsets of ''X'' are: :(I)=\min\. ::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose union is not in ''I'' any more. As any ideal is closed under finite unions, this number is always at least \aleph_0; if ''I'' is a σ-ideal, then \operatorname(I) \ge \aleph_1. :\operatorname(I)=\min\. :: The "covering number" of ''I'' is the smallest number of sets from ''I'' whose union is all of ''X''. As ''X'' it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Line
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set ''R'' with the four properties # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two distinct elements there is another); # the order on ''R'' is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and # every collection of mutually disjoint non-empty open intervals in ''R'' is countable (this is the countable chain condition for the order topology of ''R''), is ''R'' necessarily order-isomorphic to the real line R? If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Hypothesis
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set ''R'' with the four properties # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two distinct elements there is another); # the order on ''R'' is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and # every collection of mutually disjoint non-empty open intervals in ''R'' is countable (this is the countable chain condition for the order topology of ''R''), is ''R'' necessarily order-isomorphic to the real line R? If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Cardinal
In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ. It is named after the n (1894–1919). See also * Suslin representation * |