# Alexander Arhangelskii

Alexander Arhangelskii
Born 13 March 1938
Moscow, Soviet Union
Fields Mathematics
Institutions Moscow State University, Ohio University
Alma mater Moscow State University
Known for General topology

Alexander Vladimirovich Arhangelskii (Russian: Александр Владимирович Архангельский, Aleksandr Vladimirovich Arkhangelsky, born 13 March 1938 in Moscow) is a Russian mathematician. His research, comprising over 200 published papers, covers various subfields of general topology. He has done particularly important work in metrizability theory and generalized metric spaces, cardinal functions, topological function spaces and other topological groups, and special classes of topological maps. After a long and distinguished career at Moscow State University, he moved to the United States in the 1990s. In 1993 he joined the faculty of Ohio University, from which he retired in 2011.

## Biography

Arhangelskii was the son of Vladimir Alexandrovich Arhangelskii and Maria Pavlova Radimova, who divorced by the time he was four years old. He was raised in Moscow by his father. He was also close to his uncle, childless aircraft designer Alexander Arkhangelsky. In 1954, Arhangelskii entered Moscow State University, where he became a student of Pavel Alexandrov. At the end of his first year, Arhangelskii told Alexandrov that he wanted to specialize in topology.[1]

In 1959, in the thesis he wrote for his specialist degree, he introduced the concept of a network of a topological space. Now considered a fundamental topological notion, a network is a collection of subsets that is similar to a basis, without the requirement that the sets be open.[2] Also in 1959 he married Olga Constantinovna.[1]

He received his Candidate of Sciences degree (equivalent to a Ph.D.) in 1962 from the Steklov Institute of Mathematics, supervised by Alexandrov.[3] He was granted the Doctor of Sciences degree in 1966.

It was in 1969 that Arhangelskii published what is considered his most significant mathematical result. Solving a problem posed in 1923 by Alexandrov and Urysohn, he proved that a first-countable, compact Hausdorff space must have a cardinality no greater than the continuum. In fact, his theorem is much more general, giving an upper bound on the cardinality of any Hausdorff space in terms of two cardinal functions. Specifically, he showed that for any Hausdorff space X,

${\displaystyle |X|\leq 2^{\chi (X)L(X)}}$

where χ(X) is the character, and L(X) is the Lindelöf number. Chris Good referred to Arhangelskii's theorem as an "impressive result", and "a model for many other results in the field."[4] Richard Hodel has called it "perhaps the most exciting and dramatic of the difficult inequalities",[5] a "beautiful inequality", and "the most important inequality in cardinal invariants."[6]

In 1970 Arhangelskii became a full professor, still at Moscow State University. He spent 1972–75 on leave in Pakistan, teaching at the University of Islamabad under a UNESCO program.[1]

Arhangelskii took advantage of the few available opportunities to travel to mathematical conferences outside of the Soviet Union.[1] He was at a conference in Prague when the 1991 Soviet coup d'état attempt took place. Returning under very uncertain conditions, he began to seek academic opportunities in the United States.[7] In 1993 he accepted a professorship at Ohio University, where he received the Distinguished Professor Award in 2003.[8]

Arhangelskii was one of the founders of the journal Topology and its Applications, and volume 153 issue 13, July 2006, was a special issue, with most of the papers based on talks given at a special conference held at Brooklyn College 30 June–3 July 2003 in honor of his 65th birthday.

## Selected publications

### Papers

• Arkhangel'skii, A.V. (1959). "An addition theorem for the weight of sets lying in bicompacta". Doklady Akademii Nauk SSSR. 126: 239–241.
• Arhangel'skiĭ, A. (1966). "Mappings and Spaces". Russian Mathematical Surveys. 21 (4): 115–162.
• Arkhangel'skiĭ, A.V. (1969). "An approximation of the theory of dyadic compacta". Soviet Mathematics. 10: 151–154.
• Arhangel'skii, A.V. (1969). "On the cardinality of bicompacta satisfying the first axiom of countability". Soviet Mathematics. 10: 967–970.
• Arkhangelskii, A. V. (1978). "Structure and Classification of Topological Spaces and Cardinal Invariants". Russian Mathematical Surveys. 33 (6): 33–96. doi:10.1070/RM1978v033n06ABEH003884.
• Arkhangel'skii, A. V. (1980). "Some properties of radial spaces". Mathematical Notes. 27 (1): 50–54. doi:10.1007/BF01149814.
• Arkhangel'skii, A. V. (1980). "Relations among the invariants of topological groups and their subspaces". Russian Mathematical Surveys. 35 (3): 1–24. doi:10.1070/RM1980v035n03ABEH001674.
• Arkhangel'skii, A. B.; Shakhmatov, D. B. (1990). "On pointwise approximation of arbitrary functions by countable families of continuous functions". Journal of Mathematical Sciences. 50 (2): 1497–1512. doi:10.1007/BF01388512.
• Arhangel'skii, A.V. (5 June 1996). "Relative topological properties and relative topological spaces". Topology and its Applications. 70 (2–3): 87–99. doi:10.1016/0166-8641(95)00086-0.

## References

1. ^ a b c d Shenfeld, Karen (17 March 1996). "In The Neighborhood of Mathematical Space (an interview with Alexander V. Arhangelskii)". Topological Commentary. 1 (1). ISSN 1499-9226. Retrieved 18 June 2012. (reprinted from the Summer 1993 issue of The Idler)
2. ^ Sakai, Masami (2004). "Topological Spaces". In Hart, Klaus P.; Nagata, Jun-iti; Vaughan, Jerry E. Encyclopedia of General Topology. Elsevier Science. p. 5. ISBN 978-0444503558.
3. ^
4. ^ Good, Chris (2004). "The Lindelöf Property". In Hart, Klaus P.; Nagata, Jun-iti; Vaughan, Jerry E. Encyclopedia of General Topology. Elsevier Science. p. 183. ISBN 978-0444503558.
5. ^ Hodel, R. (1984). "Chapter 1: Cardinal Functions I". In Kunen, Kenneth; Vaughan, Jerry E. Handbook of Set-Theoretic Topology. Amsterdam: North-Holland Publishing Company. p. 18. ISBN 0-444-86580-2.
6. ^ Hodel, R.E. (1 July 2006). "Arhangelʹskiĭ's Solution to Alexandroff's Problem: a Survey" (PDF). Topology and its Applications. Elsevier. 153 (13): 2199–2217. doi:10.1016/j.topol.2005.04.011. ISSN 0166-8641. Retrieved 23 January 2012.
7. ^ Yetter, David (1993). "Moscow, money, and mathematics: An interview with Alexander Arhangel'skii" (PDF). Friends of Mathematics Newsletter. Kansas State University Department of Mathematics. Retrieved 18 June 2012.
8. ^ "Two Ohio University faculty members named Distinguished Professor". Outlook. Ohio University. 2 October 2003. Retrieved 18 June 2012.