Vietoris's secondary education was from 1902 to 1910 at the Benedictine Gymnasium in Melk. He received a good mathematical education at this Gymnasium but at this time he was following his father's wishes and aiming for a career in engineering. In fact when he entered the Technical University in Vienna in 1910 it was still with the intention of becoming an engineer. However, he quickly decided that he wanted to become a mathematician and even as soon as Christmas 1910 he was making efforts to study mathematics and descriptive geometry. He received excellent teaching in mathematics from Hermann Rothe, in descriptive geometry from Emil Müller, and in projective geometry from Theodor Schmid. He was particularly inspired by the lectures in projective geometry which gave him a life-long love for the topic. From the academic year 1911-12 he took courses at the University of Vienna in addition to courses at the Technical University. His lecturers at the University of Vienna included Gustav von Escherich (1849-1935), Wilhelm Wirtinger, Philipp Furtwängler, Gustav Kohn (1859-1921), Wilhelm Gross (1886-1918), and the philosopher Adolf Stöhr (1855-1921).
It was in 1912 that Vietoris heard a lecture by Wilhelm Gross on topology. Gross had been a student of Wirtinger at Vienna, then a postdoctoral student at Göttingen before returning to Vienna as a Privatdocent. In this lecture Gross described his own work extending ideas by Frigyes Riesz who had defined the concept of a mathematical continuum as an abstract set provided with a notion of an accumulation point. Gross described a system of axioms defining the notions of neighbourhood and of accumulation point, three of these axioms being due to Frigyes Riesz and the fourth due to himself. Around the same time, Vietoris had attended lectures by Hermann Rothe at the Technical University where he had introduced the notion of a manifold. This gave Vietoris the idea of using a topological approach to create a geometrical notion of a manifold. He was working on these ideas, advised by Gustav von Escherich and Wilhelm Wirtinger, when World War I broke out. Austria-Hungary declared war on Serbia on 28 July 1914 and Vietoris volunteered for service in the army in August 1914. In the following month he was badly wounded and, for a considerable time, he was unable to take part in the war as he was recovering from these wounds. After his recovery he was sent to the Italian front. Italy had joined the war on 23 May 1915 when they declared war on Austria-Hungary. The Austrians held the mountains and Italy, although holding numerical superiority, found difficulty attacking the Austrians in the mountainous terrain. Vietoris, who loved the mountains, was given the task of acting as an army mountain guide. Despite the very difficult conditions of war, Vietoris was still able to think about his research problems and it was in 1916, while being a mountain guide, that he made a significant advance. Despite the wartime conditions, he was able to publish his first paper, namely Eine besondere Erzeugungsweise der Raumkurven vierter Ordnung zweiter Art Ⓣ (1916). He was able to return to Vienna for the spring semester of 1918 and during these three months he made more progress with his research and also was able to read Felix Hausdorff's Grundzüge der Mengenlehre Ⓣ which had been published in 1914. In October 1918 the Austro-Hungarians were decisively defeated at the Vittorio Veneto and, on 4 November, Vietoris was captured by the Italians. Although hostilities ended shortly after this, Vietoris was held in captivity by the Italians until 7 August 1919.
Although Vietoris was held prisoner for nine months, he was well treated and during these months he was able to complete writing his doctoral thesis. On being released and returning to Vienna, he qualified as a Gymnasium teacher of mathematics and descriptive geometry in October 1919, and submitted his thesis Stetige Mengen Ⓣ to the University of Vienna in December 1919. He was awarded his doctorate in July 1920 and his thesis, published in the Monatshefte für Mathematik in 1921, is considered by many as his most significant contribution. Having qualified as a Gymnasium teacher, Vietoris took up a teaching position in 1919. Soon, however, he received a postcard from Gustav von Escherich congratulating him on his outstanding thesis and offering him the position of Assistant Professor at the Technical University Graz. Vietoris accepted and began teaching there in 1920. At Graz he was an assistant to Roland Weitzenböck who was an ordinary professor in number theory and the theory of invariants.
In 1922 Vietoris moved to the University of Vienna where he habilitated in the following year. He had written several papers which were published in 1922 and 1923, two of which are Bereiche zweiter Ordnung Ⓣ (1922) and Kontinua zweiter Ordnung Ⓣ (1923). The first of these he had submitted as his habilitation thesis. His habilitation was recommended by Hans Hahn. In the 1923 paper he writes:-
This work is a continuation of my research on the foundations of topology: "Stetige Mengen", Monatsh. f. Math. Phys. 31 (1921) and "Kontinua zweiter Ordnung", Monatsh. f. Math. Phys.32 (1922).Also in 1922 and 1923 his papers Über Extrema mit Nebenbedingungen Ⓣ (1922), Das stetige Deformieren topologischer Gebilde vom Standpunkt der Mengenlehre Ⓣ (1923) and Zur Geometrie ebener Massenanziehungsprobleme Ⓣ (1923) were published. The authors of  write:-
The 1920s were a heady decade for topologists, and Vienna was as good a place to be as any, with Hahn, Menger, Reidemeister, and later Hurewicz and Nöbeling around. In the general commotion many ideas emerged independently and almost simultaneously in several places. Vietoris, who always was an extremely modest person, never engaged in priority debates (quite in contrast, for instance, to his young and fiery colleague Karl Menger). But Vietoris was the first to introduce filters (which he called "wreaths") and one of the first to define compact spaces (which he called "lückenlos"), using the condition that every filter has a cluster point. He also introduced the notion of regularity, and proved that (in modern parlance) compact spaces are normal.In 1925 Vietoris spent three semesters in Amsterdam having been awarded a Rockefeller fellowship. There he participated in a seminar led by L E J Brouwer but also had Pavel Sergeevich Aleksandrov, Karl Menger, David van Dantzig and Witold Hurewicz as participants. Influenced by the algebraic ideas that were being discussed in this seminar, Vietoris began to undertake research in algebraic topology. Back in Vienna he began giving lectures on homology groups and cohomology groups which are algebraic invariants of topological spaces. Walther Mayer (1887-1948) had been appointed to the University of Vienna as a privatdocent in 1926. He attended Vietoris's lectures and, in particular, tried to solve some conjectures that Vietoris had described. Vietoris had suggested how a proof of these conjectures might be found and Mayer solved a special case in 1929 which he published in Über abstrakte Topologie Ⓣ (1929). In this paper Mayer writes:-
I was introduced to topology by my colleague Vietoris, whose lectures I attended in 1926-7 at the local university. In many talks about this area Vietoris gave me many hints for which I am very grateful.Vietoris then completed the proof of the full result for homology groups in 1930 which he published in Über die Homologiegruppen der Vereinigung zweier Komplexe Ⓣ (1930). He wrote in this paper:-
W Mayer, whom I told about the problem as well as the conjectured result and a way to its solution, has solved the question, as far as it concerns Betti numbers, in a somewhat different way in these 'Monatshefte'. In what follows, I will return to my original idea and use it for the solution in the general case.Today this result is known as the Mayer-Vietoris sequence. It provides an important method to split the computation of homology groups and cohomology groups of a topological space into a computation of these groups for subspaces of the topological space.
In 1927 Vietoris left Vienna to take up a position as associate professor at the University of Innsbruck. He was very pleased to leave the hectic life in Vienna to be close to the High Alps that he loved in Innsbruck. However, after a year he returned to Vienna becoming a full Professor at the Vienna University of Technology. In the autumn of that year he married Klara Anna Maria Riccabona von Reichenfels (1904-1935), the daughter of Rudolf Riccabona von Reichenfels and his wife Maria Burlo von Ehrwall. Two years later, in 1930, he returned to the University of Innsbruck as a full professor and he remained there for the rest of his career. Let us note here that Klara died in 1935 during the birth of their sixth child (all six were daughters). In the following year, Vietoris married her sister, Maria Josefa Vincentia Riccabona von Reichenfels (1901-2002).
Back in Innsbruck, Vietoris was drawn to the mountains that he loved so much. However, he combined pleasure with scientific work :-
Vietoris became involved with the local school of glaciologists led by the distinguished Finsterwalder. In the role of a Gletscherknecht, he carried the heavy instruments for geological measurements and set up experiments in countless scientific alpine excursions. In due time, Vietoris started publishing himself on the blockstream of the Hochebenkar, a glacier-like formation of rock débris pasted together by ice, which he had come to know like no one else. He also wrote on how to use the compass as an alpinist (rather than a sailor), on "geometry in the service of the mountaineer," and on the physics of skiing, and he held patent no. 100832 for a method of using air photographs in cartography.Germany annexed Austria in March 1938 so, when they invaded Poland in September 1939 leading to countries declaring war on Germany, Austria was again at war. Vietoris again volunteered for service as he had done in World War I, but by now he was 48 years old. He was immediately sent to Poland and, as he had been in World War I, he was wounded. He continued to serve in the military until June 1941 when he reached the age of 50. At this stage he was allowed to leave military service and resume his duties at the University of Innsbruck.
Even before World War II, Vietoris had begun to undertake research in areas different from his initial work on algebraic topology. For example he published Über die Integration gewöhnlicher Differentialgleichungen durch Iteration Ⓣ (Part 1, 1932; Part 2, 1934; Part 3, 1939). During World War II he published Zur Theorie der Integraphen Ⓣ (1942) which was reviewed by William Edward Milne who writes:-
The author makes a harmonic analysis of the difference between the true curve and the curve actually followed by the tracing point of the instrument and, after adopting suitable assumptions, concludes that the resulting error is "practically nil."He wrote further on this topic after the war. Another of his papers published during the war was Zur Kennzeichnung des Sinus und verwandter Funktionen durch Funktionalgleichungen Ⓣ (1944) in which he developed a method to introduce the sine by a functional equation. After the war he worked on a number of different areas including statistics, publishing a series of four papers entitled Vergleich unbekannter Mittelwerte auf Grund von Versuchsreihen Ⓣ between 1979 and 1982. He continued to publish papers, his last paper being the third in a series entitled Über das Vorzeichen gewisser trigonometrischer Summen Ⓣ(1994) which appeared when he was 103 years old.
For his outstanding achievements Vietoris received many awards. He was elected a corresponding member of the Austrian Academy of Sciences in 1935 and a full member in 1960. He received honorary doctorates from the Technical University of Vienna (1984) and the University of Innsbruck (1994). These were for his work:-
... on orientation in mountainous terrain by differential geometric means, the strength of the alpine ski, and the physics of block glaciers.He received the Austrian Cross of Honour for Science and Art (1973), the Grand Gold Decoration of Honour for Services to the Republic of Austria (1981), the Gold Medal of the Austrian Mathematical Society (1981), and the Order of Merit of the city of Innsbruck (1982). Vietoris was elected an honorary member of the Austrian Mathematical Society in 1965 and of the German Mathematical Society in 1992.
Let us end this biography by quoting Heinrich Reitberger's summary of Vietoris's contributions :-
Leopold Vietoris's fundamental contributions to general as well as algebraic topology, and also to other branches of the mathematical sciences, have made him immortal in the world of science. As a person, he was outstandingly humble and grateful for his well-being, which he also wished and granted his fellow humans. He devoted his spare time to his large family, religious meditation, music, and his beloved mountains. On the other hand, administrative duties were not Vietoris's favorite tasks, as he pointed out in a letter to L E J Brouwer in 1947: "As dean I am overwhelmed with administrative matters to such an extent that I often have to hold my lectures inadequately prepared and don't have any time for scientific research. Luckily, the term will soon be over and then I hope to be a scientist again and not a bureaucrat." In research Vietoris was a "lone fighter": Only one of his more than seventy mathematical papers has a coauthor. Half of the papers were written after his sixtieth birthday.
Article by: J J O'Connor and E F Robertson