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:
“One way or another we all work for our vice.”
—Ben Maddow (1909–1992)
“... actresses require protection in their art from blind abuse, from savage criticism. Their work is their religion, if they are seeking the best in their art, and to abuse that faith is to rob them, to dishonor them.”
—Nance O’Neil (1874–1965)
“To say that a work of art is good, but incomprehensible to the majority of men, is the same as saying of some kind of food that it is very good but that most people can’t eat it.”
—Leo Tolstoy (1828–1910)