Editing Felix Hausdorff (section)

===The "Magnum Opus": "Principles of set theory"===
According to previous notions, set theory included not only the general set theory and the theory of sets of points, but also dimension and measure theory. Hausdorff's textbook was the first to present all of set theory in this broad sense, systematically and with full proofs. Hausdorff was aware of how easily the human mind can err while also seeking for rigor and truth, so in the preface of his work he promises:

{{blockquote|… to be as economical as possible with the human privilege of error.}}

This book went far beyond its masterful portrayal of already-known concepts. It also contained a series of important original contributions by the author.

The first few chapters deal with the basic concepts of general set theory. In the beginning Hausdorff provides a detailed set algebra with some pioneering new concepts (differences chain, set rings and set fields, <math>\delta</math>- and <math>\sigma</math>-systems). The introductory paragraphs on sets and their connections included, for example, the modern set-theoretic notion of functions. Chapters 3 to 5 discussed the classical theory of cardinal numbers, order types and ordinals, and in the sixth chapter "Relations between ordered and well-ordered sets" Hausdorff presents, among other things, the most important results of his own research on ordered sets.

In the chapters on "point sets"—the topological chapters—Hausdorff developed for the first time, based on the known neighborhood axioms, a systematic theory of topological spaces, where in addition he added the separation axiom later named after him. This theory emerges from a comprehensive synthesis of earlier approaches of other mathematicians and Hausdorff's own reflections on the problem of space. The concepts and theorems of classical point set theory <math>\mathbb{R}^n</math> are—as far as possible—transferred to the general case, and thus become part of the newly created general or set-theoretic topology. But Hausdorff not only performed this "translation work", but he also developed basic construction methods of topology such as core formation (open core, [[limit point|self-dense core]]) and shell formation ([[closure (topology)|closure]]), and he works through the fundamental importance of the concept of an open set (called "area" by him) and of the concept of compactness introduced by Fréchet. He also founded and developed the theory of the connected set, particularly through the introduction of the terms "component" and "quasi-component".

With the first Hausdorff countability axiom, and eventually the second, the considered spaces were gradually further specialized. A large class of spaces satisfying the countable first axiom are [[metric space]]s. They were introduced in 1906 by Fréchet under the name "classes (E)". The term "metric space" comes from Hausdorff. In ''Principles'', he developed the theory of metric spaces and systematically enriched it through a series of new concepts: [[Hausdorff metric]], [[complete metric space|complete]], [[total boundedness]], <math>\rho</math>-connectivity, reducible sets. Fréchet's work is not particularly famous; only through Hausdorff's ''Principles'' did metric spaces become common knowledge to mathematicians.

The chapter on illustrations and the final chapter of ''Principles'' on measure and integration theory are enriched by the generality of the material and the originality of presentation. Hausdorff's mention of the importance of measure theory for [[probability]] had great historical effect, despite its laconic brevity. One finds in this chapter the first correct proof of the [[strong law of large numbers]] of [[Émile Borel]]. Finally, the appendix contains the single most spectacular result of the whole book, namely Hausdorff's theorem that one cannot define a volume for all bounded subsets of <math>\mathbb{R}^n</math> for <math>n \geq 3</math>. The proof is based on Hausdorff's [[Hausdorff paradox|paradoxical ball]] decomposition, whose production requires the axiom of choice.<ref>For the history of Haussdorff's sphere paradox see ''Gesammelte Werke Band&nbsp;IV'', S.&nbsp;11–18; also the article by P.&nbsp;Schreiber in Brieskorn 1996, S.&nbsp;135–148, and the monograph Wagon 1993.</ref>

During the 20th century, it became the standard to build mathematical theories on axiomatic set theory. The creation of axiomatically founded generalized theories, such as general topology, served among other things to single out the common structural core for various specific cases or regions and then set up an abstract theory, which contained all these parts as special cases. This brought a great success in the form of simplification and harmonization, and ultimately brought with itself an economy of thought. Hausdorff himself highlighted this aspect in the ''Principles''. In the topological chapter, the basic concepts are methodologically a pioneering effort, and they paved the way for the development of modern mathematics.

''Principles of set theory'' appeared in April 1914, on the eve of the First World War, which dramatically affected scientific life in Europe. Under these circumstances, the effects Hausdorff's book on mathematical thought would not be seen for five to six years after its appearance. After the war, a new generation of young researchers set forth to expand on the abundant suggestions that were included in this work. Undoubtedly, topology was the primary focus of attention. The journal ''[[Fundamenta Mathematicae]]'', founded in Poland in 1920, played a special role in the reception of Hausdorff's ideas. It was one of the first mathematical journals with special emphasis on set theory, topology, the theory of real functions, measure and integration theory, functional analysis, logic, and foundations of mathematics. Across this spectrum, a special focus was placed on topology. Hausdorff's ''Principles'' was cited in the very first volume of [[Fundamenta Mathematicae]], and through citation counting its influence continued at a remarkable rate. Of the 558 works (Hausdorff's own three works not included), which appeared in the first twenty volumes of Fundamenta Mathematicae from 1920 to 1933, 88 of them cite ''Principles''. One must also take into account the fact that, as Hausdorff's ideas became increasingly commonplace, so too were they used in a number of works that did not cite them explicitly.

The Russian topological school, founded by [[Paul Alexandroff]] and [[Paul Urysohn]], was based heavily on Hausdorff's ''Principles''. This is shown by the surviving correspondence in Hausdorff's [[Nachlass]] with Urysohn, and especially Alexandroff and Urysohn's ''Mémoire sur les multiplicités Cantoriennes'',<ref>Urysohn, P.: ''[http://matwbn.icm.edu.pl/ksiazki/fm/fm7/fm714.pdf Mémoire sur les multiplicités Cantoriennes.] (PDF; 6,2&nbsp;MB)'' Fundamenta Mathematicae 7 (1925), S.&nbsp;30–137; 8 (1926), S.&nbsp;225–351.</ref> a work the size of a book, in which Urysohn developed dimension theory and ''Principles'' is cited no fewer than 60 times.

After the Second World War there was a strong demand for Hausdorff's book, and there were three reprints at Chelsea from 1949, 1965 and 1978.