Borsuk, more my contemporary, came for a longer visit from Warsaw. We started collaborating from the first. From him I learned about the truly geometric, more visual, almost "palpable" tricks and methods of topology. Our results were published in a number of papers which we sent to Polish journals and to some journals abroad. Actually my first publication in the United States appeared while I was in Lwów. It was a joint paper with Borsuk, published in the Bulletin of the American Mathematical Society. We defined the idea of "epsilon homeomorphisms" — approximate homeomorphisms — and the behavior of some topological invariants under such more general transformation's — continuous ones, but not necessarily one to one. A joint paper on symmetric products introduced an idea that modifies the definition of a Cartesian product and leads to the construction of some curious manifolds. Some of these might one day find applications in physical theories. They correspond to the new statistics of counting the numbers of particles (not in the familiar classical sense, but rather in the spirit of quantum theory statistics of indistinguishable particles, or of particles obeying the Bose-Einstein or else Fermi-Dirac ways of counting their combinations and dispositions). These cannot be explained here; perhaps this mention will whet the curiosity of some readers.
Kuratowski and Steinhaus, each in a different way, represented elegance, rigor, and intelligence in mathematics. Kuratowski was really a representative of the Warsaw school which flourished almost explosively after 1920. He came to Lwów in 1927, preceded by a reputation for his work in pure set theory and axiomatic topology of general spaces. As editor of Fundamenta Mathematicae he organized and gave direction to much of the research in this famous journal. His mathematics was characterized by what I would call a Latin clarity. In the proliferation of mathematical definitions and interests (now even more bewildering than at that time), Kuratowski's measured choice of problems had the quality of what is hard to define — common sense in the abstractions.
Steinhaus was one of the few Polish professors of Jewish descent. He came from a well-known, quite assimilated Jewish family. A cousin of his had been a great patriot, one of the Pilsudski legionnaires; he was killed during the first World War.
Steinhaus's sense of analysis, his feelings for problems in real variables, in function theory, in orthogonal series manifested a great knowledge of historical development of mathematics and continuity of ideas. Perhaps without so much interest or feeling for the very abstract parts of mathematics, he also steered some new mathematical ideas in the direction of practical applications.
He had a talent for applying mathematical formulations to matters as common as problems of daily life. Certainly his inclinations were to single out problems of geometry that could be treated from a combinatorial point of view — actually anything that presented the visual, palpable challenge of a mathematical treatment.
He had great feeling for linguistics, almost pedantic at times. He would insist on absolutely correct language when treating mathematics or domains of science susceptible to mathematical analysis.
Auerbach was rather short, stooped, and usually walked with his head down. Outwardly timid, he was often capable of very caustic humor. His knowledge of classical mathematics was probably greater than that of most of the other professors. For example, he knew classical algebra very well.
At his instigation Mazur, a few others, and I decided to start a systematic study of Lie groups and other theories which were not strictly in the domain of what is now called Polish mathematics. Auerbach also knew a lot about geometry. I had many discussions with him on the theory of convex bodies, to which Mazur and I contributed several joint papers.
Auerbach and I played chess at the Café Roma and often went through the following little ritual when I began with a certain opening (at that time I did not know any theory of chess openings and played by intuition only). When I made those moves with the king pawn he would say, "Ah! Ruy Lopez." I would ask, "What is that?" and he would reply, "A Spanish bishop."
Auerbach died during the war. I understand that he and Sternbach took poison while being transported by the Germans to an interrogation session, but I do not know the circumstances of' their arrest or anything else about their lives before and during the Nazi occupation.
I believe my collaboration with Schreier started when I was in my second year of studies. Of the mathematicians at the University and at the Polytechnic Institute, he was the only one who was more strictly my contemporary, since he was only six months or a year older and still a student at the University. We met in a seminar room during a lecture by Steinhaus and talked about a problem on which I was working. Almost immediately we found many common interests and began to see each other regularly. A whole series of papers which we wrote jointly came from this collaboration.
We would meet almost every day, occasionally at the coffee house but more often at my house. His home was in Drohobycz, a little town and petroleum center south of Lwów. What a variety of problems and methods we discussed together! Our work, while still inspired by the methods then current in Lwów, branched into new fields: groups of topological transformations, groups of permutations, pure set theory, general algebra. I believe that some of our papers were among the first to show applications to a wider class of mathematical objects of modern set theoretical methods combined with a more algebraic point of view. We started work on the theory of groupoids, as we called them, or semi-groups, as they are called now. Several of these results can be found in the literature by now, but some others have not yet appeared in print anywhere to my knowledge.
Schreier was murdered by the Germans in Drohobycz in April, 1943.
Another mathematician, Mark Kac, four or five years my junior, was a student of Steinhaus. As a beginning undergraduate he had already shown exceptional talent. My connections with Kac developed a little later during my summer visits to Lwów, when I began to spend academic years at Harvard. He also had the good fortune to come to the United States, a few years after I did, and our friendship started in full measure only in this country.
In 1932 I was invited to give a short communication at the International Mathematical Congress in Zürich. This was the first big international meeting I attended, and I felt very proud to have been invited. In contrast to some of the Polish mathematicians I knew, who were terribly impressed by western science, I had confidence in the equal value of Polish mathematics. Actually this confidence extended to my own work. Von Neumann once told my wife, Françoise, that he had never met anyone with as much self-confidence — adding that perhaps it was somewhat justified.
Traveling west, I first joined Kuratowski, Sierpinski, and Knaster in Vienna. They had all come from Kuratowski's summer place near Warsaw; on the way to Zürich the professors decided to stop in Innsbrück. We met some mathematicians from other countries also on their way to the Congress and spent a couple of days there. I remember an excursion by cablecar to a mountain called Hafelekar. This was the first time I was ever above two thousand meters, and the view was beautiful. I remember feeling a little dizzy for a few minutes and identifying this feeling with one I had had previously on several occasions when getting the salient points of proofs of theorems I studied in high school.