# Eduard Čech

### Quick Info

Stracov, Bohemia (now Czech Republic)

Prague, Czechoslovakia (now Czech Republic)

**Eduard Čech**was a Czech mathematician known for his work in projective differential geometry and topology. He introduced the idea of the higher homotopy groups of a space.

### Biography

**Eduard Čech**'s father was Čenek Čech, a policemen, and his mother was Anna Kleplova. Eduard, his parent's fourth child, was born in Stracov which was in northeastern Bohemia but he attended the Gymnasium in Hradec Kralove where his talent for mathematics first became evident and he decided at this time that he wanted to be a school teacher of mathematics.

He entered the Philosophy Faculty of Charles University of Prague in 1912, with the aim of becoming a school teacher. He had little interest in physics so he choose courses which were within the area of mathematics and descriptive geometry. Already Čech was studying mathematical texts which were much more advanced than the level of his courses and he spent much time in the Mathematics library.

His studies were interrupted by World War I and, although he was able to complete three years of study, he was drafted into the Austro-Hungarian army in 1915. Čech spent three years in the army, but they were not totally wasted as far as his academic studies were concerned for he took the opportunity to learn Italian, German and Russian. When the war ended in 1918 he was able to return to the Charles University and complete the degree which allowed him to teach mathematics in schools.

Having obtained his degree, Čech began to teach in secondary schools in Prague but he continued to undertake research in mathematics and completed his doctorate in 1920. His interest in a new area of mathematics, namely projective differential geometry, led to his first paper on the topic appearing in 1921 and, in the same year, he obtained a scholarship from the Ministry of Education to go to Italy and study with Fubini in Turin. Čech studied there between 1921 and 1922 and he clearly impressed Fubini who asked him to cooperate with him in joint project to write a monograph on projective differential geometry. This joint work with Fubini appeared in two volumes, the first in 1926 and the second in the following year. The collaboration was clearly a highly successful one for they cooperated again in writing another major text which appeared in 1931.

After leaving Italy in 1922, Čech wrote his habilitation thesis, becoming a docent at the Charles University of Prague. This post did not carry a salary, so Čech continued to teach in the secondary schools of Prague to earn enough money to live. Tomás Masaryk was the political leader who liberated the Czechs and Slovaks from Austrian rule and in 1918 he was elected president of Czechoslovakia. A year after the founding of Czechoslovakia, a new university, named the Masaryk University after the first president, was founded in Brno. Mathias Lerch became the first professor of mathematics at this, the second Czech university, when it opened in 1920. However, Lerch died in 1922 and the opportunity arose for Čech to gain a chair and a permanent university post. Čech was appointed extraordinary professor at Masaryk University in 1923. There he lectured on analysis and algebra, becoming a full professor in 1928.

Čech was interested in geometry but he was appointed to the chair of analysis at Masaryk University since this had been Lerch's chair. There was another professor of mathematics at Masaryk University, namely Seifert, who held the geometry chair. So [14]:-

... although geometry was Čech's field of research, Čech had to take over courses in analysis and algebra. He proceeded to master these two fields.Through his widening mathematical interests, to some extent forced on him by his teaching duties, Čech became interested in topology, in particular he became one of the foremost experts on combinatorial topology. The topology papers written by Polish mathematicians in the new journal

*Fundamenta mathematicae*greatly excited him. His early interests in topology were in homology theory, a topic on which he published in 1932, and he proved duality theorems for manifolds. His aim was to bring together point-set topology and algebraic topology with his 1932 paper. In this paper, which introduces the topic which today is called the Čech homology theory, Čech also introduced the notion of the inverse limit.

At the International Congress of Mathematicians in Zürich in 1932, Čech introduced the notion of higher homotopy groups of a space. These were independently investigated by Hurewicz. Two years later Čech extended his work on homology to local homology. On hearing Čech talk about his results at a combinatorial topology conference in Moscow, Lefschetz invited him to visit Princeton and Čech made the visit during session 1935-36.

Čech was influenced by the work of Aleksandrov and Urysohn and he set up a topology seminar at Brno in 1936 which went on to produce 26 papers in 3 years. One of these papers was Čech's paper

*On bicompact spaces*(1937) which introduced what today is called the Stone-Čech compactification of regular topological spaces.

*On bicompact spaces*investigated certain topological spaces which Tikhonov had introduced in 1930 and [1]:-

Čech's interpretation became a very important tool of general topology and also of some branches of functional analysis.Čech's Brno seminar only ended when the Czech universities were closed down at the start of World War II. Čech, however, tried to continue the work of his seminar despite the War and it continued to meet at the flat of B Pospisl, one of his students, until 1941. At this point Pospisl was arrested by the Gastapo and the seminar could no longer continue. In fact Pospisl was released by the Nazis from prison in 1944 but died shortly afterwards.

Despite the difficulties which Čech encountered during World War II, he did spend much time working on his book Topological spaces which he later rewrote and published in 1959. In 1945, after the end of World War II, Čech returned to the Charles University of Prague and began an administrative career. Before the start of the war Čech had begun to organise courses for school teachers in Brno. He believed passionately in the improvement of mathematics teaching at all levels and decided to try to improve the way mathematics was taught in schools. He continued this interest after the war ended and used his experiences with the school teachers to organise a series of school mathematics textbooks. He was involved in school reforms and chaired a commission whose remit was to devise a new syllabus for school level mathematics.

Čech became Director of the Mathematical Research Institute of the Czech Academy of Sciences in 1947, Director of the Central Mathematical Institute in 1950, and Director of the Czech Academy in 1952. However in that year he returned to the Charles University of Prague to head the new Mathematical Institute there. In the 1950s his mathematical interests turned to differential geometry and after a gap through the war years he began to publish again, writing 17 papers on that topic. In 1956 he was appointed as the first director of the Mathematical Institute of the Charles University of Prague. However, his health began to fail but [14]:-

... when already gravely ill, he performed two further important services for Czech mathematics. He founded the journal Commentationes Mathematicae Universitatis Carolinae, the first issue appeared in 1960, and he came up with the idea of organising in Prague an international topological conference. The conference took place in 1961 under the name Symposium on general topology and its relations to Modern Analysis and Algebra. Since then, every five years there has been a Prague Topological Symposium.Let us end this biography by giving some details of Čech's character. Koutnik writes in [14]:-

Whenever he was doing something in mathematics, he always strove to achieve a thorough understanding of the subject. The result was that even outside his fields of research he had an extensive knowledge and deep insight into many other areas of mathematics. This feature of his personality also had some other consequences. While he was not conceited and talked easily to people with little formal education, he expected in his fellow professors the same qualities that he himself possessed. this did not contribute to smooth relations with some people as he was not diplomatic but, on the contrary, quite forthright in expressing his opinions.

### References (show)

- J Folta, L Novy, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - M Katetov and P Simon (eds.),
*The Mathematical legacy of Eduard Cech*(Basel, 1993). - P S Aleksandrov and S P Finikov, Eduard Cech: Obituary (Russian),
*Uspekhi matematicheskikh nauk***16**(1) (1961),119-126. - B Balcar, V Koutnik and P Simon, Eduard Cech, 1893-1960,
*Czechoslovak Math. J.***43**(118) (3) (1993), 567-575. - B Balcar, V Koutnik and P Simon, Eduard Cech (1893-1960),
*Acta Univ. Carolin. Math. Phys.***34**(2) (1993), 5-6. - B Balcar, V Koutnik and P Simon, Eduard Cech: 1893-1960,
*Math. Slovaca***43**(3) (1993), 381-392. - M Cenkl, Eduard Cech,
*The Cech centennial, Contemp. Math.***181**(Providence, R.I., 1995), x-xi. - Z Frolik, The personality of Eduard Cech (thoughts on the occasion of his unattained eightieth birthday) (Czech),
*Pokroky Mat. Fyz. Astronom.***18**(1973), 237-247. - J J Gray, Eduard Cech,
*The Mathematical Intelligencer***16**(4) (1994), 48-49. - M Katetov, J Novak and A Svec,
*Akademik Eduard Cech, Casopis pro pestovani matematiky***85**(1966), 477-491. - I Kolár, J Rosicky and F Sik, 100th anniversary of birthday of Eduard Cech,
*Arch. Math. (Brno)***29**(1-2) (1993), i-ii. - I Kolár, 100th anniversary of birthday of Eduard Cech,
*Geometry and physics, Rend. Circ. Mat. Palermo*(2) Suppl.**37**(1994), 13-14. - I Kolar, Thoughts on Eduard Cech's work in differential geometry (Czech),
*Pokroky Mat. Fyz. Astronom.***25**(6) (1980), 306-312. - V Koutnik, Eduard Cech, 1893-1960,
*European Mathematical Society Newsletter***8**(1993), 5-7.

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Written by J J O'Connor and E F Robertson

Last Update February 2000

Last Update February 2000