Abstract
In this chapter we will abstract certain properties of a metric space and thus create a new structure that occurs frequently in mathematics. Then we will extend to this new structure the notions of a continuous function, compactness, and connectedness. As we progress through this chapter we will frequently use metric spaces to illustrate the new concepts, and readers are encouraged to pursue such issues on their own.
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Notes
- 1.
Felix Hausdorff was born in 1868 in Breslau, Germany, which became Wrocław, Poland after World War II. When Hausdorff was young, his family moved to Leipzig, where he grew up and was educated. His original interests were in literature and music, but bowing to family pressure he studied astronomy and obtained a doctorate in 1891. He published four papers in the subject and obtained a habilitation in 1895. Nevertheless, he continued to pursue his interests in literature and music. He published his first literary work in 1897 under the name Paul Mongré. Books followed in philosophy in 1898 and poetry in 1900. Clearly he also worked in mathematics because in 1902 he was appointed to an extraordinary professorship of mathematics at Leipzig and turned down the offer of a similar appointment at Göttingen. He continued with literature and published a farce in 1904 that was produced and was an apparent success. After 1904, however, his efforts shifted to topology, introducing the concept of a partially ordered set (§ A.4). In 1910 he went to Bonn, and in 1913 he accepted an ordinary professorship at Greifswalf. (Hausdorff came from a rich family and had no financial worries, hence the willingness to accept a lower position.) In 1914 he published his book Grundzüge der Mengenlehre, in which he set out the theory of topological spaces, building on the work of Fréchet. In a sense, this is the start of point set topology, and the reader can find here the introduction of what we are calling the Hausdorff property. The book was reprinted several times and is a good place to practice your mathematical German. He continued to be active until 1935 when, as a Jew, he was forced by the Nazis to retire. He continued to do research but could not find an outlet for his work in Germany. He tried to emigrate in 1939 but was unsuccessful. In 1941 he was scheduled to be sent to a concentration camp, but he managed to avoid this. Bonn University requested that he and his wife be allowed to remain in their home, and this was granted. He committed suicide together with his wife and her sister in 1942.
- 2.
James Waddell Alexander was born in 1888 in Sea Bright, New Jersey. His father was the American painter John White Alexander. In 1915 he received his doctorate from Princeton University, having previously spent time studying mathematics in Bologna and Paris. He married in 1917. During World War I he entered the army as a lieutenant and at the end of the war left as a captain. He returned to Princeton as an Assistant Professor in 1920 and was promoted in 1928 to Professor. From 1933 on he was a member of the Institute for Advanced Study. During World War II he was a civilian working with the U.S. Army Air Force at their Office of Scientific Research and Development. Because of his leftist political views he came under the scrutiny of the McCarthy Committee; this had the effect of turning him into a recluse after his retirement in 1951. His research focused on topology, particularly algebraic topology, in which he was a pioneer of cohomology theory. His named contributions include the Alexander Duality Theorem, the Alexander horned sphere, the present result, and the Alexander polynomial used in knot theory. There is also Alexander–Spanier cohomology theory, which he introduced in 1935 and that was generalized to its present form by Spanier in 1948. In addition, he impressed those who knew him as a charming man with a fondness for limericks and mountain climbing. His climbing was centered in the Swiss Alps and Colorado Rockies. Alexander’s Chimney, in the Rocky Mountain National Park, is named after him. He died in 1971 in Princeton, New Jersey.
- 3.
Andrei Nikolaevich Tikhonov was born in 1908 in Smolensk, Russia. (His name is often written Tychonoff.) He entered Moscow State University in 1922 and published his first paper in 1925 while still an undergraduate. He received his doctorate in 1927 and was appointed to the faculty of the university in 1933. He first proved the present theorem for the product of an arbitrary infinity of copies of the unit interval and in 1935 stated the full result with the comment that the proof was the same as in the special case. In 1936 he received his habilitation for work on Volterra functional equations and then was made Professor at Moscow State University. Three years later he became a Corresponding Member of the USSR Academy of Sciences. His work now concentrated on differential equations and mathematical physics. In 1966 he was awarded the Lenin Prize and was elected to full membership in the Soviet Academy of Sciences. He had a long and distinguished career including administrative positions as Dean of the Faculty of Computing and Cybernetics at Moscow State University and later as Deputy Director of the Institute of Applied Mathematics of the USSR Academy of Sciences. He died in 1993.
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Conway, J.B. (2014). Topological Spaces. In: A Course in Point Set Topology. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-02368-7_2
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