Lev Abramovich Tumarkin (; 14 January 1904 – 1 August 1974) was a Soviet

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who made significant contributions to

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, particularly in

dimension theory In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...

. He served as dean of the Faculty of Mechanics and Mathematics at

Moscow State University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...

from 1935 to 1939.

Biography

Tumarkin was born in

Hadiach Hadiach (, ) is a List of cities in Ukraine, city in Myrhorod Raion, Poltava Oblast in east-central Ukraine. It hosts the administration of . Hadiach is located on the Psel (river), Psel River. Population: Name In addition to the Ukrainian lan ...

(then part of the

Russian Empire The Russian Empire was an empire that spanned most of northern Eurasia from its establishment in November 1721 until the proclamation of the Russian Republic in September 1917. At its height in the late 19th century, it covered about , roughl ...

Poltava Governorate Poltava Governorate was an administrative-territorial unit (''guberniya'') of the Russian Empire. It was officially created in 1802 from the disbanded Little Russia Governorate (1796–1802), Little Russia Governorate and had its capital in Polt ...

). He graduated from Moscow State University in 1925 and completed his postgraduate studies there in 1929 under the supervision of

Pavel Alexandrov Pavel Sergeyevich Alexandrov (), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote roughly three hundred papers, making important contributions to set theory and topology. In topol ...

. He spent his entire academic career at Moscow State University, where he became a professor in 1932 and earned his

doctorate A doctorate (from Latin ''doctor'', meaning "teacher") or doctoral degree is a postgraduate academic degree awarded by universities and some other educational institutions, derived from the ancient formalism '' licentia docendi'' ("licence to teach ...

in physical and

mathematical sciences The Mathematical Sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper. Statisti ...

in 1936.

Mathematical contributions

Tumarkin began his research career early, making notable contributions to topology while still an undergraduate. His main work focused on

. * Between 1925 and 1928, Tumarkin proved that for topological spaces with countable base, the large and small

inductive dimension In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ' ...

s are equal:

\mathrm\,X = \mathrm\,X

* He showed that any n-dimensional space with countable base can be represented as a union of

n+1

pairwise disjoint zero-dimensional sets * Hurewicz–Tumarkin theorem (1927): Every n-dimensional

compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

contains an n-dimensional Cantor manifold (proved independently by

Witold Hurewicz Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician who worked in topology. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on th ...

) * Tumarkin's theorem (1928): For any subset

M

of a space

X

with countable base, there exists a set

M'

that is a union of

countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

closed sets 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. In a complete metric space, a ...

X

such that

M = M'

and

\dim M' = \dim M

* In 1951, he proved that the

weight In science and engineering, the weight of an object is a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition. Some sta ...

of any one-dimensional compact space equals either two or three. * In 1957, he demonstrated that every infinite-dimensional compact space either contains an infinite-dimensional Cantor manifold or contains compact sets of every finite dimension.

Tumarkin's problem

In 1925, Tumarkin posed the following problem:

Tumarkin's problem: ''Does there exist an
infinite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
where every non-empty
closed subset 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. In a complete metric space, a cl ...
has dimension either
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
or infinity?''

The question remained open for over 40 years until

American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, p ...

mathematician

David W. Henderson David Wilson Henderson (February 23, 1939 – December 20, 2018) was a Professor Emeritus of Mathematics in the Department of Mathematics at Cornell University. His work ranges from the study of topology, algebraic geometry, history of mathemati ...

provided a positive answer in 1967, showing that such "Tumarkin compacts" form a

dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...

in the space of all infinite-dimensional compact sets.

Soviet The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...

mathematicians

and

Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet ...

described his teaching as "the fruit of many years of creative work and finished with filigree thoroughness." Mathematician

Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...

, one of Tumarkin's

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

students, praised his teaching on Arnold's website.

Publications

* "Zur allgemeinen Dimensionstheorie" (1925) (in German) * "Über die Dimension night abgeschlossener Mengen" (1928) (in German) * "О покрытиях одномерных компактов" (1951) (in Russian) * "О бесконечномерных канторовых многообразиях" (1957) (in Russian) * "О сильно- и слабо-бесконечномерных пространствах" (1963) (in Russian)

References

{{DEFAULTSORT:Tumarkin, Lev 1904 births 1974 deaths 20th-century Russian mathematicians Topologists

Biography

Mathematical contributions

Tumarkin's problem

Publications

See also

References