Work
Hausdorff was the first to state a generalization of Cantor's Continuum Hypothesis; his Aleph Hypothesis, which appears in his 1908 article Grundzüge einer Theorie der geordneten Mengen, and which is equivalent to what is now called the Generalized Continuum Hypothesis.
In 1909, while studying partially ordered sets of real sequences, he stated what is now known as the Hausdorff Maximal Principle; he was the first to apply a maximal principle in algebra.
In his 1914 classic text, Grundzüge der Mengenlehre, he defined and studied partially ordered sets abstractly; using the Axiom of Choice, he proved that every partially ordered set has a maximal linearly ordered subset. In this same book, he axiomatized the topological concept of neighborhood and introduced the topological spaces that are now called Hausdorff spaces.
In 1914 using the Axiom of Choice, he gave a "paradoxical" decomposition of the 2-sphere as the disjoint union of four sets A, B, C, and Q, where Q is countable and the sets A, B, C, and B ∪ C are mutually congruent. This later inspired the Banach–Tarski paradoxical decomposition of the ball in 3-space.
He introduced the concepts now called Hausdorff measure and Hausdorff dimension, which have been useful in the theory of fractals. In analysis, he solved what is now called the Hausdorff moment problem. In addition, Hausdorff spaces are named after him, as is the Hausdorff distance on the collection of nonempty closed subsets of a metric space.
Hausdorff also published philosophical and literary works under the pseudonym "Paul Mongré". "Paul Mongre" published a number of books and articles on the philosopher Friedrich Nietzsche, as well as a number of reviews of contemporary literature and drama. Mongre-Hausdorff also published a satirical play which performed in a dozen German cities. In the course of attempts to refute Nietzsche's doctrine of "the eternal return of the same," Hausdorff was led to Cantor's set theory, which set Hausdorff on the road to his set-theoretical discoveries. Hausdorff's Nietzschean philosophical writings appear in volume VII of his collected works.
A project to publish Hausdorff's works and biography, along with a description of his mathematical contributions, in nine volumes, is underway, edited by E. Brieskorn, F. Hirzebruch, W. Purkert, R. Remmert, E. Scholz.
Read more about this topic: Felix Hausdorff
Famous quotes containing the word work:
“If work is part of your identity, think very carefully before you give it up. Giving it up won’t make you a better mother; it will make you less of the person you are; and that will make you less of a mother.”
—Jean Marzollo (20th century)
“Men should not labor foolishly like brutes, but the brain and the body should always, or as much as possible, work and rest together, and then the work will be of such a kind that when the body is hungry the brain will be hungry also, and the same food will suffice for both; otherwise the food which repairs the waste energy of the overwrought body will oppress the sedentary brain, and the degenerate scholar will come to esteem all food vulgar, and all getting a living drudgery.”
—Henry David Thoreau (1817–1862)
“A work of art is one through which the consciousness of the artist is able to give its emotions to anyone who is prepared to receive them. There is no such thing as bad art.”
—Muriel Rukeyser (1913–1980)