{{Infobox scientist | name = Lev Abramovich Tumarkin | image = Lev Tumarkin Moscow, 1935.tiff | image_size = | caption = Lev Tumarkin at the First Topological Conference, Moscow, 1935 | birth_date = {{birth date|1904|1|14}} | birth_place = Hadiach, Poltava Governorate, Russian Empire | death_date = {{death date and age|1974|8|1|1904|1|14}} | death_place = Moscow, Soviet Union | field = Mathematics | workplaces = Moscow State University | alma_mater = Moscow State University (1925) | doctoral_advisor = Pavel Alexandrov | known_for = Dimension theory<br>Tumarkin's theorem<br>Tumarkin's problem | academic_advisors = }}
'''Lev Abramovich Tumarkin''' ({{Langx|ru|Лев Абра́мович Тума́ркин}}; 14 January 1904 – 1 August 1974) was a Soviet mathematician who made significant contributions to topology, particularly in dimension theory. He served as dean of the Faculty of Mechanics and Mathematics at Moscow State University from 1935 to 1939.<ref name="mathmech">{{cite book |title=Mathematicians and Mechanics - Rectors of Moscow University and Deans of the Faculty of Mechanics and Mathematics |editor=V. N. Chubarikov |year=2004 |publisher=Faculty of Mechanics and Mathematics Publishing |pages=73–74 |language=ru}}</ref>
== Biography == Tumarkin was born in Hadiach (then part of the Russian Empire's Poltava Governorate). He graduated from Moscow State University in 1925 and completed his postgraduate studies there in 1929 under the supervision of Pavel Alexandrov. He spent his entire academic career at Moscow State University, where he became a professor in 1932 and earned his doctorate in physical and mathematical sciences in 1936.<ref name="math40">{{cite book |title=Mathematics in the USSR for Forty Years. 1917-1957 |volume=2 |editor=A. G. Kurosh |year=1959 |publisher=Fizmatgiz |pages=690 |language=ru}}</ref><ref name="mech80">{{cite book |title=MSU Faculty of Mechanics and Mathematics 80: Mathematics and Mechanics at Moscow University |editor=A. T. Fomenko |year=2013 |publisher=Moscow University Press |pages=298 |isbn=978-5-19-010857-6 |language=ru}}</ref>
== Mathematical contributions == Tumarkin began his research career early, making notable contributions to topology while still an undergraduate.<ref name="mathmech2">{{cite book |title=Mathematicians and Mechanics - Rectors of Moscow University and Deans of the Faculty of Mechanics and Mathematics |editor=V. N. Chubarikov |year=2004 |publisher=Faculty of Mechanics and Mathematics Publishing |pages=81 |language=ru}}</ref> His main work focused on dimension theory.
* Between 1925 and 1928, Tumarkin proved that for topological spaces with countable base, the large and small inductive dimensions are equal: <math>\mathrm{Ind}\,X = \mathrm{ind}\,X</math><ref name="alex">{{cite book |author1=P. S. Alexandrov |author2=V. A. Pasynkov |title=Introduction to Dimension Theory |year=1973 |publisher=Nauka |pages=187, 275–277, 344 |language=ru}}</ref> * He showed that any n-dimensional space with countable base can be represented as a union of <math>n+1</math> pairwise disjoint zero-dimensional sets<ref name="alex"/> * '''Hurewicz–Tumarkin theorem''' (1927): Every n-dimensional compact space contains an n-dimensional Cantor manifold (proved independently by Witold Hurewicz)<ref name="alex"/> * '''Tumarkin's theorem''' (1928): For any subset <math>M</math> of a space <math>X</math> with countable base, there exists a set <math>M'</math> that is a union of countably many closed sets in <math>X</math> such that <math>M = M'</math> and <math>\dim M' = \dim M</math><ref>{{cite book |author1=P. S. Alexandrov |author2=V. A. Pasynkov |title=Introduction to Dimension Theory |year=1973 |publisher=Nauka |page=385 |language=ru}}</ref> * In 1951, he proved that the weight of any one-dimensional compact space equals either two or three.<ref>{{cite journal |author=L. A. Tumarkin |title=On coverings of one-dimensional compacta |journal=Moscow University Mathematics Bulletin |year=1951 |issue=3 |pages=3–14 |language=ru}}</ref> * 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.<ref>{{cite journal |author=L. A. Tumarkin |title=On infinite-dimensional Cantor manifolds |journal=Doklady Akademii Nauk SSSR |volume=115 |year=1957 |pages=244–246 |language=ru}}</ref>
=== Tumarkin's problem === In 1925, Tumarkin posed the following problem: <blockquote>'''Tumarkin's problem:''' ''Does there exist an infinite-dimensional compact set where every non-empty closed subset has dimension either zero or infinity?''</blockquote> The question remained open for over 40 years until American mathematician David W. Henderson provided a positive answer in 1967, showing that such "Tumarkin compacts" form a dense set in the space of all infinite-dimensional compact sets.<ref>{{cite book |author1=P. S. Alexandrov |author2=V. A. Pasynkov |title=Introduction to Dimension Theory |year=1973 |publisher=Nauka |page=493 |language=ru}}</ref><ref name="g144">{{cite journal | last=Henderson | first=David W. | title=An Infinite-Dimensional Compactum with no Positive-Dimensional Compact Subsets-A Simpler Construction | journal=American Journal of Mathematics | publisher=Johns Hopkins University Press | volume=89 | issue=1 | year=1967 | issn=0002-9327 | jstor=2373100 | pages=105–121 | doi=10.2307/2373100 | url=http://www.jstor.org/stable/2373100 | access-date=2025-01-12| url-access=subscription }}</ref>
Soviet mathematicians Pavel Alexandrov and Andrey Kolmogorov described his teaching as "the fruit of many years of creative work and finished with filigree thoroughness."<ref>{{cite journal |author1=P. S. Alexandrov |author2=A. N. Kolmogorov |title=Lev Abramovich Tumarkin (On his 60th birthday) |journal=Russian Mathematical Surveys |volume=19 |issue=4 |year=1964 |pages=219–221 |doi=10.1070/RM1964v019n04ABEH001157 |bibcode=1964RuMaS..19..161A |language=ru |url=http://www.mathnet.ru/links/01f2d9ad99d8e68485c83ab10c123578/rm6236.pdf}}</ref><ref>{{cite book |author=V. B. Demidovich |title=On the History of the Faculty of Mechanics and Mathematics, Moscow State University |year=2013 |publisher=Publishing House of the Board of Trustees of the Faculty of Mechanics and Mathematics, Moscow State University |pages=322 |isbn=978-5-211-01978-2 |language=ru}}</ref> Mathematician Vladimir Arnold, one of Tumarkin's calculus students, praised his teaching on Arnold's website.<ref>{{cite web | url=https://www.math.fsu.edu/~wxm/Arnold.htm | title=V.I. Arnold, on teaching mathematics }}</ref>
== 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)
== See also == * Dimension theory
==References== {{reflist}}
{{Authority control}} {{DEFAULTSORT:Tumarkin, Lev}} Category:1904 births Category:1974 deaths Category:20th-century Russian mathematicians Category:Topologists