Commentary on Menger’s Work on Dimension Theory

Zusammenfassung Karl Menger (1902–1985) initially began investigating mathematical ideas concerning the concepts of curve and dimension during his first months as a student at the University of Vienna. He had entered the university in the autumn of 1920. Near the beginning of 1921 Hans Hahn (1879–1934) arrived in Vienna as professor of mathematics and in the spring organized a seminar on the topic ‘Neueres über den Kurvenbegriff’(‘New [ideas] concerning the concept of curve’), covering recent research on curves and dimension. The first session of the seminar occurred in late April or early May. Menger attended at the urging of his friend Otto Schreier (1901–1929). For additional historical details see Johnson [1981: 233–240] and Sigmund [1998: 7]. At this time numerous open problems and recent results motivated mathematicians to investigate the concepts of curve and dimension. Many unusual, even paradoxical examples and results surrounded these concepts. A few years earlier L. E. J. Brouwer (1881–1966) had satisfactorily proved the topological invari–ance of dimension and had proposed a mathematical definition of dimension (see Brouwer [1911, 1913]), though Menger did not at first know about these results. For detailed historical information on the origins and development of dimension theory see Johnson [1977] [1979] [1981], Reichel [1980], Menger [1994: Chapter IV], Katětov and Simon [1997], Koetsier and Mill [1997], and Crilly [1999].