After receiving his doctorate from the University of Budapest Kerékjártó took up a scholarship abroad. In the summer semester of 1922 at the invitation of the University of Gšttingen he gave a course on topology and the following semester a course titled "Mathematische Betrachtungen zur Kosmologie" [Mathematical considerations in Cosmology]. The first of these courses was enlarged in his book Vorlesungen über Topologie [Lectures on Topology] which appeared in the series "Grundlehren der Mathematischen Wissenschaften". This work was the first of its kind and inspired much later research in this young branch of geometry. The following year (1923) he gave courses at the University of Barcelona titled "Geometry and the theory of functions". He spent the years 1923/4 and 1924/5 as a visiting lecturer at Princeton where he gave courses on topology and on continuous groups. He returned from the United States at the invitation of the Sorbonne. By his "Lessons on topology and its applications" he gained the respect of the most distinguished French geometers. His friendship with foreign mathematicians continued to deepen for the rest of his life.
Returning to Hungary in 1926, he became Professor of Geometry first at the University of Szeged and then (in 1938) at the University of Budapest. However, his cooperation with the rest of the mathematical world continued and he was often invited by other universities and academies. In his addresses at international congresses he always considered the most important problems of topology.
His most important scientific results were in the area of "classical" topology founded by Poincaré and Brouwer and in the theory of continuous groups.
After the problem of classifying surfaces up to homeomorphism had been solved (and that for open surfaces resolved by Kerékjártó) it became possible to study more deeply the structure of the transformations of such surfaces. At the beginning of Kerékjártó's research there were two classic results in this area. The first was "Poincaré's last geometric theorem", later proved by G D Birkhoff, and the second was "Brouwer's translation theorem". Kerékjártó had earlier shown, in his book, the close relationship between these two theorems, and in an article in these Proceedings in 1928, he showed how they could have a common proof. This demonstration of the simultaneous proof of the two results united some widely separated areas of mathematics. In fact, while the first theorem played an important role in Poincaré's research in dynamics, the second was applied by Brouwer in the theory of continuous groups. Also in his later work he preferred to work in topological problems which were closely connected with problems of classical geometry, theory of functions, etc.
The famous speech by Hilbert at the International Congress in Paris had been of great importance to the development of topology. It had also given impetus to the development of the theory of continuous groups. The most beautiful results of this theory, in the case of dimension 2, are due to Kerékjártó. It is enough to mention the topological characterisation of the homographic representations of the sphere and of the affine group of the plane, the foundations of complex projective geometry and theorems on the transitive groups of the line. It was these methods which also led to fundamental results on topology and Euclidean and hyperbolic geometry in 3 dimensions.
He also studied the regular transformations of surfaces, which are closely connected to the above problems and which lead to interesting applications in dynamics.
His expertise in this new branch of geometry, topology, was recognised in, among other ways, his being asked to write the chapter on Topology in the Encyclopédie Francaise.
Kerékjártó was always very careful in his choice of courses to teach at university. He introduced his students to many important areas of geometry.
It is a well-balanced sense of order that also shows itself in his final work: a book of more than 600 pages, in Hungarian, dealing with projective geometry from its beginnings to the most modern problems. In spite of the richness of the subject-matter and the rigour of the treatment it is still an enjoyable read.
This book was, in fact, the second volume in a proposed series of five. In the first, which appeared in 1937, he gave an axiomatic foundation, with no missing elements, for metric Euclidean geometry. The following volumes would have dealt with the other areas of geometry. It is a great loss to mathematics that these volumes cannot appear.
Death has taken Béla de Kerékjártó when he was still at his most creative. We do not know what plans he had in mind, but certainly his death has deprived mathematics of some really irreplaceable works.