Witold attended elementary school in Łódź in a Russian controlled Poland but with World War I beginning before he had begun secondary school, major changes occurred in Poland. At the outbreak of war, the Hurewicz family left Łódź and travelled to Moscow where Witold attended secondary school from 1914 to February 1919. In August 1915 the Russian forces which had held Poland for many years withdrew. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly a strong school of mathematics grew up in the University of Warsaw, with topology being one of the main topics. The Hurewicz family returned to Łódź in February 1919 and Witold completed his secondary school studies at the Oswiata Gymnasium in that city. He passed the matriculation examination in May 1921 and graduated from the Oswiata Gymnasium. Although Hurewicz knew by this time that he wanted to specialise in mathematics and fully understood that there was a vigorous school of mathematics in Poland, nevertheless he chose to go to Vienna to continue his studies. He left Łódź to travel to Vienna on 16 July 1921.
In Vienna, he studied under Hans Hahn, receiving a Ph.D. in 1926 for his thesis Über eine Verallgemeinerung des Borelschen Theorems Ⓣ. Karol Borsuk explains how the Vienna school of mathematics determined the direction of Hurewicz's research :-
At that time Vienna was a very prosperous place for mathematics, and besides Hahn (who authored the excellent exposition 'Theorie der reellen Funktionen' Ⓣ), there were also many other outstanding mathematicians, including N Hofreiter and W Wirtinger. Under Hahn's powerful influence, an active mathematics research centre of set theory formed in Vienna. Among the many eminent set theorists there were Karl Menger, one of the creators of dimension theory, and his colleagues Kurt Gödel (who consequently became famous as one of the most thorough investigators of the principles of set theory), Georg Nöbeling, and Abraham Wald. In this environment Hurewicz turned to set theory.Although Borsuk gives a good indication of Hahn's Vienna School in this quote, it is a little confusing since not all those mentioned were teaching there when Hurewicz was a student. Wilhelm Wirtinger was appointed to a chair at the University of Vienna in 1905 and Hahn was appointed in 1921. Karl Menger entered the University of Vienna in 1920 to study physics but changed to write a doctoral thesis on dimension advised by Hahn which he completed in 1924. Nikolaus Hofreiter (1904-1990) studied from 1923 in Vienna with Hans Hahn, Wilhelm Wirtinger, Emil Müller and Philipp Furtwängler. He was a student at the same time as Hurewicz and was awarded his doctorate in 1927. Abraham Wald entered the University of Vienna in 1927 to study with Karl Menger and was awarded his doctorate in 1929.
Hurewicz began to work on extending theorems of Menger and Urysohn on dimension, which they had proved for Euclidean spaces, to separable metric spaces. To do this he had to produce new techniques and he began to publish his results in a series of papers. The first few are: Über schnitte von Punktmengen Ⓣ (1926), Stetige bilder von Punktmengen Ⓣ (1926), Grundiss der Mengerschen Dimensionstheorie Ⓣ (1927), Normalbereiche und Dimensionstheorie Ⓣ (1927), Stetige bilder von Punktmengen. II Ⓣ (1927), Verhalten separabler Räume zu kompakten Räumen Ⓣ (1927), and Über Folgen stetiger Funktion Ⓣ (1927). As Borsuk writes that, through these papers :-
... Hurewicz became known as one of the creators of dimension theory, next to Menger and Urysohn.He was awarded a Rockefeller scholarship which allowed him to spend the year 1927-28 in Amsterdam. He remained in Amsterdam being appointed as a docent and an assistant to L E J Brouwer from 1928 to 1936. Samuel Eilenberg writes about this period in :-
I first met Hurewicz when I was a student at the University of Warsaw. It was around 1932-1933. To me he was an idol, a Jew from Poland who became a prominent world mathematician in a field I was in love with: an ideal to admire and to follow. Hurewicz was then in Holland and came to Warsaw almost once a year. We talked about mathematics, and I discussed what I was doing. He was supportive and helpful. Once when I proved something good, I wrote to him and received a very congratulatory reply. I still have that letter. At the same time I met Lefschetz who visited Warsaw on several occasions. We all three met in Oslo in 1936 on the occasion of the International Mathematical Congress. At that time my future was discussed, and it was agreed that I should visit Western Europe first (Paris, Zürich, Oxford, and Cambridge) before moving to America. In the fall of 1936 I started implementing this plan and went to Paris for a six-month stay. I was helped in various ways by Professor Wacław Sierpiński. At the time Hurewicz was already in America.He was given study leave for a year which he decided to spend in the United States. In September 1936 he sailed from Rotterdam to New York on the Statendam. He visited the Institute for Advanced Study in Princeton and spent the year 1936-37 as a fellow there. He decided to remain in the United States and not return to his position in Amsterdam but he came back to Europe in the summer of 1937, returning from Le Havre to New York on the SS President Roosevelt in October 1937. Given the impending war in Europe this was clearly a wise decision. He returned to the Institute for Advanced Study in Princeton and, in January 1938, he applied for citizenship of the United States. At this time he was living at 1 Evelyn Place, Princeton. He describes himself as having brown hair, brown eyes, height 5 ft 6 in, weight 145 lbs, of Hebrew race and Polish nationality. He remained at the Institute until 1939 although he again visited Europe in the summer of 1938, returning to New York from Le Havre in September of that year on the Champlain. In April 1939 he went with his colleague Henry Wallman to meet his friend Samuel Eilenberg when he arrived in New York on the SS Manhattan. Eilenberg writes :-
I arrived to New York on April 27, 1939, and there at the pier were Hurewicz and Wallman to take me by car for about ten days to Princeton, which was then the undisputed mathematical mecca of the world. On the way we stopped for a snack, and I was introduced to cinnamon toast, which just became a big fad. I was also introduced to car trouble, as the lights refused to work when we were ready to continue.After his years at Princeton, Hurewicz was appointed first to the University of North Carolina being an assistant professor there from 1939 to 1942. His parents, Mieczyslaw and Katarzyna Hurewicz, sailed to New York from Rotterdam on the Niew-Amsterdam in April 1939. They returned to Europe but sailed from Genoa, Italy to New York in June 1940 on the SS Manhattan. His brother Stefan also emigrated to the United States via the Philippines in January 1941. This route via the Philippines was a common one for Jews fleeing Nazi persecution. Hurewicz was registered for the draft on 16 February 1942. During World War II he contributed to the war effort with research on applied mathematics, in particular the work he did on servomechanisms at that time was classified because of its military importance. He was still officially on the staff of the University of North Carolina from 1942 to 1945 although he was given leave of absence for government service. In fact he was promoted to Associate Professor in 1942. He did have time to accept the position of Visiting Professor at Brown University, Providence, Rhode Island from January 1943 to June 1944 living at 1 Megee, Providence. He gave a series of lectures at Brown University in 1943 and these were published in mimeographed form by Brown University as Ordinary differential equations in the real domain with emphasis on geometric method. The notes covered existence theorems, linear systems, and geometrical aspects of non-linear systems in the plane. During 1944-45 he worked at the Radiation Laboratory at the Massachusetts Institute of Technology in Cambridge, Massachusetts.
From 1945 until his death he worked at the Massachusetts Institute of Technology. He was an Associate Professor of Mathematics there from 1945 to 1948 when he was promoted to full professor. He lived at 993 Memorial Drive, Cambridge, Massachusetts with his mother who was now calling herself Catherine. In September 1950 he attended the International Congress of Mathematicians at Cambridge, Massachusetts and delivered the invited plenary lecture Homology and Homotopy in the Topology Section of the Congress. During the summer of 1953 he was in Paris, lecturing on 'Homotopy' at the Collège de France, then flying back from Paris to Boston on 7 August on Air France. During the autumn of 1953 he again visited the Institute for Advanced Study in Princeton. He also spent the summer of 1954 visiting Europe, attending the International Congress of Mathematicians in Amsterdam from 2 September to 9 September, then visiting Paris from where he flew back to Boston on 16 September on Air France. Hurewicz died falling off a ziggurat (a Mexican pyramid) while on a conference outing at the 'International Symposium on Algebraic Topology' in Mexico. His fall was on the 4 September 1956 and he was taken to the medical centre in Mérida, Mexico with serious injuries, dying two days later. The death certificate, signed by Dr Fernando Guzmán-Espinoso, gives the cause of death as "traumatic and hemorrhagic shock resulting from severe fractures." In  it is suggested that his legendary absentmindedness was a factor:-
Hurewicz, who never married, was a highly cultured and charming man, and a paragon of absentmindedness, a failing that probably led to his death.Eilenberg writes :-
When Hurewicz died in Mérida after the Mexico City conference of 1956, I and several other participants were still in Mexico City. I remember sitting with a group of friends in the gardens of a hotel (a converted cloisterlike seventeenth-century hospital) when the news arrived. It was a black day.At his brother Stefan's request, Hurewicz was cremated and his ashes were shipped back to Mount Auburn Cemetery, Cambridge, Massachusetts. Rather strangely, it is noted on his death certificate that, "Neither a United States passport nor any other evidence of citizenship of the deceased was found among his personal effects."
As we have already noted, Hurewicz's early work was on set theory and topology and :-
... a remarkable result of this first period  is his topological embedding of separable metric spaces into compact spaces of the same (finite) dimension.In the field of general topology his contributions are centred around dimension theory. He wrote, in collaboration with Henry Wallman, an important text Dimension theory published in 1941. The authors write in the Preface:-
In this book it has been the aim of the authors to give a connected and simple account of the most essential parts of dimension theory. Only those topics were chosen which are of interest to the general worker in mathematics as well as the specialist in topology. Since the appearance of Karl Menger's well-known 'Dimensions theorie' in 1928, there have occurred important advances in the theory, both in content and in method. These advances justify a new treatment, and in the present book great emphasis has been laid on the modern techniques of function spaces and mappings on spheres. The algebraically minded reader will find in Chapter VIII a concise exposition of modern homology theory, with applications to dimension. Historical references are made solely for the guidance of the beginning student, and no attempt has been made to attain completeness in this respect.A reviewer writes that the book:-
... is truly a classic. It presents the theory of dimension for separable metric spaces with what seems to be an impossible mixture of depth, clarity, precision, succinctness, and comprehensiveness.Karol Borsuk writes :-
... for separable metric spaces, the book by Hurewicz and Wallman remains a model of clarity and strict parallels between the theory and the geometric insight. Their book also contains algebraic topology notions and methods introduced to dimension theory by P S Aleksandrov.In addition to this book, Hurewicz is best remembered for two remarkable contributions to mathematics, his discovery of the higher homotopy groups in 1935-36, and his discovery of exact sequences in 1941. His work led to homological algebra. It was during Hurewicz's time as Brouwer's assistant in Amsterdam that he did the work on the higher homotopy groups :-
... the idea was not new, but until Hurewicz nobody had pursued it as it should have been. Investigators did not expect much new information from groups, which were obviously commutative ...Hurewicz had a second textbook published, but this was not until 1958 after his death. Lectures on ordinary differential equations was a reprinting, with minor revisions, of the mimeographed notes of his Brown University lectures. Perhaps it is worth noting that the mimeographed notes had been reissued by the Mathematics department of the Massachusetts Institute of Technology in 1956. This textbook is a beautiful introduction to ordinary differential equations which again reflects the clarity of his thinking and the quality of his writing.
Let us end our biography of Hurewicz by quoting from the Preface to  written by Krystyna Kuperberg:-
Witold Hurewicz is known for his contributions to dimension theory, algebraic topology (mainly, the 1930s papers on homotopy theory and the work on fibrations), and applied mathematics. Among his published works are two excellent books ... Both books, beautifully written in a clear and concise manner so characteristic of Hurewicz, are continuously popular, and have been reprinted more than once. ...
Article by: J J O'Connor and E F Robertson