# Vitaly Vitalievich Fedorchuk

Born
1 November 1942
Moscow, Russia
Died
9 December 2012
Moscow, Russia

### Biography

It is claimed in [3] that Vitaly Vitalievich Fedorchuk was the son of Vitaly Vasilyevich Fedorchuk. Vitaly Vasilyevich Fedorchuk (1918-2008) was born into a family of farmers in the Ukraine, studied at a military school, and served in Ukraine with Smersh, the military counterintelligence service, from 1943 to 1947. At this time he became known as "the butcher of the Ukraine". He later served as chairman of the K.G.B. for a short time in 1982 and was then Interior Minister from 1982 to 1986. We are unable to confirm from any independent sources that the mathematician Vitaly Vitalievich Fedorchuk, the subject of this biography, was his son. There are several obituaries of Vitaly Vasilyevich Fedorchuk (see for example [2]) but none of these tell us he had a son who became a mathematician. However, given that Vitaly Vasilyevich Fedorchuk worked for Smersh and the K.G.B. it is quite believable that his son's existence would be kept a secret.

Vitaly Fedorchuk, the subject of this biography, entered Moscow State University in 1959. He completed his undergraduate course in 1964 and, after this, remained at Moscow State University, undertaking research advised by Pavel Sergeevich Aleksandrov. He was awarded his Candidate's Degree (equivalent to a Ph.D.) in 1967 for his thesis Perfect Irreducible Mappings of Topological Spaces and q-proximities (Russian). Fedorchuk began research while an undergraduate and published On w-mappings of paracompact spaces (Russian) in 1963. In this paper he gave variations on Hugh Dowker and Arthur Harold Stone's results concerning conditions under which a Hausdorff space is paracompact. While undertaking research, Fedorchuk published Ordered sets and the product of topological spaces (Russian) (1966) and Ordered spaces (Russian) (1967). He published a short paper giving results from his thesis in 1967 which was reviewed by D V Thampuran who writes:-
In this paper the author introduces the concept of a q-space, which is a generalization of proximity spaces. A q-space is defined on a regular space by a binary relation q which satisfies conditions similar to those of a proximity space. A q-space is consistent with a completely regular space. The relationship between q-spaces on a regular space X and bicompact extensions of all its perfect irreducible inverse mappings is given.
This paper only contained announcements of the results and Fedorchuk published the proofs in his paper Perfect irreducible mappings and generalized proximities (Russian) (1968). The paper Ordered spaces (Russian) (1967) which we mentioned above, also only announced results and proofs of the theorems stated in that paper appeared in Ordered proximity spaces (Russian) (1968).

Fedorchuk worked at Moscow State University and in 1977 he defended his doctoral thesis (equivalent in standard to the habilitation or D.Sc.) Inverse spectra of topological spaces and some problems of general topology related to dimension and cardinal functions. He became head of the General Topology and Geometry department at Moscow State University in 1982 and, beginning in the following year, he also ran P S Aleksandrov's research seminar on general topology. An overview of his research contributions is given in [1]:-
The scientific work of Vitaly Fedorchuk covers various areas of topology and has found applications in functional analysis, differential geometry, measure theory and probability theory. Fedorchuk's constructions and concepts have become important tools of research. His main scientific interests were general topology, manifolds and cell complexes, and algebraic topology. Fedorchuk is the author of 173 scientific papers and books on various aspects of dimension theory, uniform topology, extensors, the theory of covariant functors, the theory of infinite-dimensional spaces and manifolds.
He co-authored a number of books written at various different levels. With Boris A Pasynkov he published Topology and dimension theory (Russian) (1984). Although this short 63-page booklet was written jointly, in fact the two authors wrote different parts of the text. Fedorchuk wrote a survey of the basic notions and theorems of the theory of metric spaces (presented in Section 2) and topological spaces (presented in Sections 4 and 5), as well as a survey of dimension theory (presented in Section 6). The rest of the book was written by Pasynkov as an introduction to Fedorchuk's sections. In 1989 he produced another booklet, this time co-authored with Vladimir V Filippov, entitled Topology of hyperspaces and its applications (Russian). Despite the subject appearing to be quite advanced, the authors aimed the booklet to be:-
... within the reach not only of beginning university students, but also of older secondary school students interested in mathematics.
Reviewing the booklet, Roman Duda writes:-
It starts leisurely with the notion of topological metric spaces and their hyperspaces consisting of closed subsets (with the Hausdorff distance or Vietoris topology), but then gathers momentum as it runs through function spaces (metric of uniform convergence, compact-open topology, linear spaces, norm), exponential functors (including parts on connectedness and symmetric products), multivalued mappings (with semicontinuities, selections, retractions), probability measures, spaces of partial mappings, axiomatization of solution spaces for an ordinary differential equation (with a digression on optimal control), and autonomous spaces. The contents are extensive and logically nicely ordered, and the language is clear and precise. However, the reader's attention is turned only to the conceptual build-up, while proofs and references are completely omitted. As a result, a mature mathematician can perhaps read this text with some pleasure as a sort of a survey, but one may doubt what profit can be gained by a beginner reading about, e.g., the Riesz theorem on a measure representation of a functional, or the Poincaré-Bendixson theorem on curves without self-intersections - without much explanation and no indication of where to turn for further details.
Finally, we mention his research monograph Absolute retracts and infinite-dimensional manifolds (Russian) (1992) co-authored with Alex Chigogidze. This book surveys all the developments in infinite-dimensional manifolds from the time the theory began to be studied in the 1960s up to the time that the book was written:-
The authors often discuss proofs, emphasizing strategy and methods, and in many cases provide technical details. They also discuss the sources and give a bibliography of over 200 entries.
Jan van Mill, who was working at the Vrije University in Amsterdam, describes an interesting interaction with Fedorchuk in [3]. Van Mill received an email from Fedorchuk in the spring of 1999 asking if he could send him a proof of an assertion that appeared in the 1941 book Dimension Theory by Witold Hurewicz and Henry Wallman. After attempting a proof and having failed, van Mill replied to Fedorchuk explaining he had tried but failed to prove the assertion. Fedorchuk wrote back saying he was in a similar position. After further exchanges of emails in which they told each other about finding the same assertion in several books and papers but always without a proof, Fedorchuk asked van Mill if he could arrange for him to visit Amsterdam where they could work together on the problem. The visit was arranged for 1 to 15 May 1999. Van Mill writes:-
A few weeks before his visit, I began to think seriously about the problem. And when I picked him up from Schiphol Airport on 1 May 1999, I was in a state of great excitement. When we saw each other, I said to him before shaking hands: "Vitaly, I can do it!" He replied: "Jan, I can do it too". ... On 11 May 1999 Vitaly spoke at the topology seminar at the University of Amsterdam on dimension theory. Two weeks later, I talked about our joint success that we later published in 'Fundamenta Mathematicae'.
Their joint paper was entitled Dimensionsgrad for locally connected Polish spaces and it was published in 2000. The authors summary is:-
It is shown that for every $n ≥ 2$ there exists an $n$-dimensional locally connected Polish space with Dimensionsgrad 1.
We note that 'Dimensionsgrad' was a dimension function due to L E J Brouwer which was based on ideas of Henri Poincaré. In their paper, Fedorchuk and van Mill write:-
In 1913, Brouwer presented the first definition of a dimensional invariant intended for Polish spaces without isolated points (in the terminology of his days: normal sets in the sense of Fréchet). He called it "Dimensionsgrad".
In this paper Fedorchuk thanked:-
... the Division of Mathematics and Computer Science of Vrije Universiteit for generous hospitality and support.
Let is end our biography by quoting the authors of [1] who write:-
Fedorchuk devoted much time and energy to the organization of the educational and scientific processes at Moscow State University, being a member of the Academic Board of the university and the Academic Council of the Mechanics and Mathematics Faculty. In 2006, Vitaly Vitalievich Fedorchuk was awarded the title of Honorary Professor at Moscow State University. From 2008 until his death he was the director of the Baku branch of Moscow State University.

### References (show)

1. S D liadis, K L Kozlov and J van Mill, Vitaly Vitalievich Fedorchuk (1942-2012), Topology and its Applications 179 (2015), 2-4.
2. D Martin, Vitaly Fedorchuk, 89, of K.G.B., Dies, The New York Times (9 March 2008).
3. J van Mill, Brouwers dimensionsgrad: controverse en verwarring, Nieuw Archief voor Wiskunde 5/14 (2013), 130-138.