# Johannes de Groot

### Quick Info

Garrelsweer, The Netherlands

Rotterdam, The Netherlands

**Johannes de Groot**was a Dutch mathematician who worked in topology.

### Biography

**Johannes de Groot**'s mother was Maria Margaretha Kuylman (1881-1969) and his father, also named Johannes de Groot (1886-1942), was, at the time his son was born, a minister in a church. In 1928 Johannes de Groot Sr. was appointed professor of Hebrew linguistics and Israelite archeology at the University of Groningen, succeeding Franz M Th Bohl. In 1931 he became professor of Exegesis of the Old Testament at the University of Utrecht. He married Maria Margaretha Kuylman who was the daughter of a minister of the Church and had literary gifts; she wrote two novels.

The de Groot family had been traditionally a farming family. In the 17th century they had settled in the Het Bildt region which had been reclaimed from the Middelsee (or Bordine). The father of Johannes de Groot Sr. had owned a large farm on the border of Groningen and Friesland. However, falling transportation costs saw an American 'grain invasion' into Europe around 1890 which had a massive impact on the European farmers. The grandfather of the subject of this biography gave up farming and lived in Groningen on the income from his properties. Despite being a farming family, the de Groot family was fascinated with mathematical problems and, when there was little farming activity during the winter months, they would entertain themselves solving mathematical problems. De Groot Sr. was interested in mathematics but had decided to study theology and literature at the University of Groningen.

Johannes de Groot, the subject of this biography, attended the Christelijk Gymnasium in The Hague for two years and Willem Lodewijk Gymnasium in Groningen for two years. It was his mathematics teacher J Scholten at the Willem Lodewijk Gymnasium who first awakened his interest in mathematics which led to him deciding to study mathematics at university. He entered the University of Groningen in 1933 but, although mathematics was his main subject, he also studied physics and philosophy as secondary subjects. Among his mathematics lecturers were Johannes Gaultherus van der Corput (1890-1975) and Gerrit Schaake (1892-1945). Van der Corput's area of research was analytic number theory and he was a professor at Groningen from 1923 to 1946. He then moved to Amsterdam where he was the cofounder, with David van Dantzig and Jurjen Ferdinand Koksma, of the research and service institution, the Mathematisch Centrum, in February 1946. Schaake had studied at Amsterdam and, advised by Hendrik de Vries, was awarded his Ph.D. in 1922 for the thesis

*Afbeeldingen van figuren op de punten eener lineaire ruimte*Ⓣ. After graduating with his first degree, de Groot went on to study for his doctorate, advised by Schaake. He began to undertake research on algebraic geometry also looking at certain problems in algebra. However, he moved towards topology and was awarded a Ph.D. in 1942 for the 102-page thesis entitled

*Topologische Studien. Compactificatie, Voortzetting van Afbeeldingen en Samenhang*Ⓣ. He had already published a summary of the results of his thesis in the 6-page paper written in German entitled

*Sätze über topologische Erweiterung von Abbildungen*Ⓣ which appeared in 1941. Baayen and Maurice write in [4]:-

In his thesis, de Groot considered several interconnected problems concerning compactification of topological spaces, extension of mappings and (quasi-)connectivity.Many fascinating conjectures arose from the work in de Groot's thesis which are discussed in the book [2]. We quote below from the Preface of that book. Hans Freudenthal met de Groot's while de Groot was working on his thesis, see [5] or [6]:-

I got to know Johannes de Groot in the summer of 1941 when he was working on his thesis. When he came to me, he was already well advanced and in possession of the main result ... Indeed, he had chosen that topic independently - in all his studies he has been his own mentor. I recognized the work which he described to me was related to the subject of my dissertation and the work of Leo Zippin - his work partly overlapped ours, although his angle of approach was quite different.Of course, de Groot's years studying at university had been made difficult by World War II. This began in September 1939 with the German invasion of Poland but at this stage The Netherlands remained neutral. Their neutrality was not respected, however, for in May 1940 the German armies took over the country. The Netherlands was under German occupation for the following years and this was certainly the case when de Groot completed his university studies. It was not until May 1945 that Allied forces liberated the country. The position in 1942 was that although under German occupation, the country's administration attempted to keep things operating as best as they could in very difficult circumstances. De Groot became a secondary school teacher of mathematics, first at Coevorden and then at The Hague. However, he continued his mathematical research and over the following years published a number of papers. Let us look at some papers he published in 1942, the year his Ph.D. was awarded.

In 1942 he published

*On the extension of continuous functions*in which he proves that it is impossible to extend every continuous (bounded) real function on a nonclosed subset of a metric space to a continuous (bounded) real function on that space. In

*Bemerkung zum Problem der topologischen Erweiterung von Abbildungen*Ⓣ (1942) he proves that given a nonclosed or a nonbounded subset of $\mathbb{R}^{n}$ then there exists a homeomorphism taking the given subset into a subset of $\mathbb{R}^{n}$ with the property that this homeomorphism can not be extended to $\mathbb{R}^{n}$. In another 1942 paper,

*Bemerkung über die analytische Fortsetzung in bewerteten Körpern*Ⓣ, he gives a new proof of the theorem that it is impossible to obtain an analytic extension of a power series with coefficients in a $p$-adic field. Nathan Jacobson reviewed de Groot's 1942 paper

*Topologische Eigenschaften bewerteter Körper*Ⓣ and writes:-

An element x of an extension of a non-Archimedean valued field is called completely transcendental if it is not the limit of a sequence of algebraic elements. As is well known, any field can be obtained from its prime field by a succession of simple transcendental extensions followed by a succession of algebraic extensions. The authors show that a field with a nontrivial non-Archimedean valuation is separable if and only if it is possible to choose the (transfinite) sequence of transcendental extensions in such a way that there is only a denumerable number of elements which are completely transcendental over the field that they extend. Using a result of Zippin (1935) it is proved that the space of such a field can be made compact by adjoining a denumerable number of points.In 1946, with the country beginning to recover from World War II, de Groot was appointed as a scientific officer at the Mathematical Centre in Amsterdam. There were two universities in Amsterdam, the University of Amsterdam (founded 1632) and the Free (Vrije) University (founded 1880). The Mathematical Centre, however, was an independent institution not attached to either of these universities. We mentioned above that Van der Corput, one of de Groot's influential teachers at university, had been the co-founder of the Mathematical Centre in Amsterdam in February 1946.

The following year, de Groot was appointed a lecturer in mathematics at the University of Amsterdam. Then, in 1948, he was appointed professor of mathematics at the Technological University of Delft. De Groot married Lutgerdina Steffina Koster in 1949; they had one daughter. Three years later, in 1952, he was appointed Professor of Mathematics at the University of Amsterdam. He also retained his position at the Mathematical Centre in Amsterdam and, in 1960, was appointed Head of Pure Mathematics there.

In 1964 he became Dean of the Faculty of Science at the University of Amsterdam and, at this time, he gave up his position of Head of Pure Mathematics at the Mathematical Centre but remained associated with the Mathematical Centre as Advisor to Pure Mathematics [4]:-

... actively participating in and in many instances decisively influencing its research activity.Although he worked in Amsterdam for the rest of his career, he made many visits to the United States. He visited Purdue University (September 1959-September 1960), Washington University at St. Louis (September 1963-February 1964), the University of Florida at Gainesville (August 1966-March 1967), and the University of South Florida in Tampa (December 1971-March 1972). In fact, from November 1967 he was appointed graduate research professor at the University of Florida at Gainesville and, from that time on, divided his time between this position and his professorship in Amsterdam.

We mentioned above that before de Groot began researching in topology he was interested in algebra. In fact, later in his career he returned to his interest in algebra when he undertook some research in group theory. One of the topics he studied was that of rigid groups, that is groups with only trivial automorphisms. These play a role in

*Automorphism groups of rings*which was the topic of his address to the International Congress of Mathematicians in Edinburgh, Scotland, in 1958.

Later de Groot worked on set-theoretic topology. He introduced the concept of co-compactness and other topological concepts. Baayen writes in [1]:-

In the last ten years of his life, de Groot was involved with new general approaches, mainly in set-theoretic topology. His early work in compactness and its generalizations, combined with new interest in Baire spaces, led him to introduce concepts like subcompactness, co-compactness, and co-topological properties in general. He elaborated a new compactification method through the use of so-called superextensions. He introduced and studied topological operators that change a given topology into another one by assigning as a closed subbase for the new topology all sets in the first topology with a specific property (for instance, all compact sets, or all connected closed sets). In his last few years, de Groot also began to do research in infinite dimensional topology and in the topology of manifolds.The book by J M Aarts and T Nishiura,

*Dimension and extensions*(1993), discusses a long-standing problem of de Groot. The main conjecture made by him, which went back to his 1942 thesis, had been solved not long before this book was written. In the Preface of this book the problem of de Groot is described as follows:-

Two types of seemingly unrelated extension problems are discussed in this book. Their common focus is a long-standing problem of Johannes de Groot, the main conjecture of which was recently resolved. As is true of many important conjectures, a wide range of mathematical investigations had developed. These investigations have been grouped into the two extension problems under discussion. The problem of de Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension. This minimal dimension was called the compactness deficiency of a space. Early success in 1942 lead de Groot to invent a generalization of the dimension function, called the compactness degree of a space, with the hope that this function would internally characterize the compactness deficiency which is a topological invariant of a space that is externally defined by means of compact extensions of a space. From this, the two extension problems were spawned. The first extension problem concerns the extending of spaces. Among the various extensions studied here, the important ones are compactifications and metrizable completions. ... The natural problems are the construction of dimension preserving extensions satisfying various extra conditions and the construction of extensions possessing adjoined subsets satisfying certain restrictions. The second extension problem concerns extending the theory of dimension by replacing the empty space with other spaces. The compactness degree of de Groot was defined by the replacement of the empty space with compact spaces in the initial step of the definition of the small inductive dimension.Hans Freudenthal explains in [5] (or [6]) the aspects of de Groot's personality which defined the direction of his research:-

He has always - as I said earlier - been his own mentor and the creator of the methods he used. Therefore, he continued to work in fields and research problems where no specific research methods existed or were required - and not where one must first pass a long route through someone else's domain before one can get started oneself. This trait was his strength, where the riches came from his grasp of simple ideas without much background knowledge, which made it possible for him to lead others to work together and to encourage them. The number of publications by others on which he has exerted influence is great.An important feature of de Groot's career was the number of Ph.D. students that he supervised who later became university lecturers. McDowell writes in [9]:-

His students essentially constitute the topology faculties at the Dutch universities.De Groot received many honours, perhaps the most prestigious of which was his election in 1969 to the Royal Dutch Academy of Sciences.

For many years de Groot suffered prolonged periods of poor health but, despite this, his death was sudden and unexpected.

### References (show)

- P C Baayen, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990).

See THIS LINK. - J M Aarts and T Nishiura,
*Dimension and extensions*(Elsevier, 1993). - P C Baayen, The topological works of J de Groot, in
*Topological Structures - Proceedings of a Conference in honour of Johannes de Groot (1914-1972)*(Math. Centrum, Amsterdam, 1974), 1-28. - P C Baayen and M A Maurice, Johannes de Groot 1914-1972,
*General Topology and its Applications***3**(1) (1973), 3-32. - H Freudenthal, Levensbericht van Johannes de Groot, J
*aarboek Koninklijke Nederlandse Akademie van Wetenschappen 1972*(1972), 119-121. - H Freudenthal, Levensbericht van Johannes de Groot,
*Koninklijke Nederlandsche Akademie van Wetenschappen*. http://www.dwc.knaw.nl/DL/levensberichten/PE00000548.pdf - Johannes De Groot (Dutch),
*Nederl. Akad. Wetensch. Verslag Afd. Natuurk.***81**(1972), 94. - L C A Leeuwen, Some Problems and Results in Abelian Group Theory,
*Niuw archief voor wiskunde***22**(2) (1974), 143-155. - R H McDowell, The works of J de Groot, in
*TOPO 72 - general topology and its applications, Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972, Lecture Notes in Math.***378**(Springer, Berlin, 1974), 1-15.

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Written by J J O'Connor and E F Robertson

Last Update October 2016

Last Update October 2016