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**Collected works. Vol. III: Set theory (1927, 1935). Descriptive set theory and topology. Edited by U. Felgner, H. Herrlich, M. Hušek, V. Kanovei, P. Koepke, G. Preuß, W. Purkert und E. Scholz.
(Gesammelte Werke. Band III: Mengenlehre (1927, 1935). Deskriptive Mengenlehre und Topologie.)**
*(German)*
Zbl 1149.01022

Berlin: Springer (ISBN 978-3-540-76806-7). xxii, 1005 p. (2008).

This volume III of Hausdorff’s (1868–1942) Collected Works (for reviews of published volumes see: Vol. II, Zbl 1010.01031, Vol. IV, Zbl 0980.01026, Vol. V, Zbl 1130.01018) contains, in its four main parts, a reprint of Mengenlehre (editions 1927; JFM 53.0169.01, 1935; JFM 61.0060.02 and Zbl 0012.20302), reprints of papers on descriptive set theory and topology, and a vast collection of excerpts from his unpublished legacy on descriptive set theory and topology – all supplied with remarks and extensive commentaries. The volume starts, as do all other in this series, with the editors’ preface and the list of all Hausdorff published papers (those reprinted here are noted with an asterisk), and it is completed by the index of names and index of subjects.

Part I of this volume consists of a historical introduction to Mengenlehre, a reprint of that book (based upon the 1927 edition, but with annotated changes from the 1935 edition), editorial remarks to the book, Hausdorff’s own remarks, reviews of the edition 1927 (list of 12, reprints of 5) and of the edition 1935 (list of 8, reprints of 2). In spite of the subtitle “second, newly reworked edition” of the Grundzüge der Mengenlehre (see volume 11 of this series, JFM 45.0123.01), Mengenlehre is a book essentially different from that. It is not only much shorter (307 pages against 476 pages, deleted are measure theory and theory of integral, and topology is not general but metric), but also contains new material (results on Zermelo’s axioms of set theory and, in modern terms, on descriptive set theory). Historical introduction (by W. Purkert, section on Lusin by V. Kanovei, altogether 40 pages long) explains differences between the two books, describes ideas leading Hausdorff to Mengenlehre, comments on abandonment by Hausdorff of the general topological framework, elaborated by Hausdorff himself, in favour of the metric one, recalls some opinions on its reception, reminds interrelations between Lusin and Hausdorff, adds some details on the 1935 edition and Russian 1937 translation. Editorial remarks (mostly by V. Kanovei, W. Purkert, 56 pages long) are of different character; they touch specific places in the book which they explain from the present point of view. Reprinted reviews were written by T. R. Bachiller, H. M. Gehman, H. Hahn, A. Rosenthal, G. T. Whyburn (edition 1927) and by Th. Skolem, G. Vivanti (edition 1935). Altogether, Part I is a fine, readable presentation of a once important book.

Part II contains reprints of all 11 Hausdorff’s published papers on descriptive set theory and topology, each followed by a commentary by one of the editors. It starts with Hausdorff’s most important contribution to descriptive set theory (continuum hypothesis holds for Borel sets, 1916; JFM 46.0291.02), his well known \(G_\delta\)-theorem in topology (each \(G_\delta\)-set in a complete metric space is homeomorphic to a complete metric space, 1924; JFM 50.0141.01), two papers on extensions of continuous mappings (1930; JFM 56.0508.03, 1938; Zbl 0018.27704 and JFM 64.0621.03), improvements of earlier results by Sierpiński, Mazurkiewicz, Kuratowski and some others.

Parts III and IV present some excerpts from Hausdorff’s unpublished legacy. The whole posthumous legacy is too voluminous to publish: it consists of 26 000 pages, of which 1000 are concerned with descriptive set theory and topology. But it is also too interesting to ignore it altogether, and so Part III contains a selection on descriptive set theory divided into six thematic sections: \(\delta s\)-operations (analytic operations); systems of sets, Borel sets, separation; Borel functions; reducible sets; Suslin sets, indices, separation; varia. Excerpts from a particular fascicle are accompanied by editorial remarks and each section is completed by a comprehensive commentary.

Part IV contains a selection on topology. It starts with three fascicles strongly related to Hausdorff’s published papers and so left without comments. These are followed by four remarkable notes on metric spaces, supplied with comments. And they in turn are followed by four very interesting Hausdorff’s essays on basic topological constructions, on curves, arcs and Peano continua, on dimension theory, and on combinatorial topology. The essays are interesting in themselves but now even more as a historical source by offering Hausdorff’s view upon the development of specific areas. The last essay (on combinatorial topology) is preceded by a valuable article by E. Scholz “Hausdorff’s view upon the beginning of algebraic topology”. It should be noted that many Hausdorff notes, now in the legacy, were written during hopelessly darkening period in his life, with no chance for publication; e.g., his proof of the topological invariance of homology groups was completed in June 1940.

Carefully edited, as are the other volumes in this series, this Volume III is an equally important contribution to the history of 20th century mathematics.

Part I of this volume consists of a historical introduction to Mengenlehre, a reprint of that book (based upon the 1927 edition, but with annotated changes from the 1935 edition), editorial remarks to the book, Hausdorff’s own remarks, reviews of the edition 1927 (list of 12, reprints of 5) and of the edition 1935 (list of 8, reprints of 2). In spite of the subtitle “second, newly reworked edition” of the Grundzüge der Mengenlehre (see volume 11 of this series, JFM 45.0123.01), Mengenlehre is a book essentially different from that. It is not only much shorter (307 pages against 476 pages, deleted are measure theory and theory of integral, and topology is not general but metric), but also contains new material (results on Zermelo’s axioms of set theory and, in modern terms, on descriptive set theory). Historical introduction (by W. Purkert, section on Lusin by V. Kanovei, altogether 40 pages long) explains differences between the two books, describes ideas leading Hausdorff to Mengenlehre, comments on abandonment by Hausdorff of the general topological framework, elaborated by Hausdorff himself, in favour of the metric one, recalls some opinions on its reception, reminds interrelations between Lusin and Hausdorff, adds some details on the 1935 edition and Russian 1937 translation. Editorial remarks (mostly by V. Kanovei, W. Purkert, 56 pages long) are of different character; they touch specific places in the book which they explain from the present point of view. Reprinted reviews were written by T. R. Bachiller, H. M. Gehman, H. Hahn, A. Rosenthal, G. T. Whyburn (edition 1927) and by Th. Skolem, G. Vivanti (edition 1935). Altogether, Part I is a fine, readable presentation of a once important book.

Part II contains reprints of all 11 Hausdorff’s published papers on descriptive set theory and topology, each followed by a commentary by one of the editors. It starts with Hausdorff’s most important contribution to descriptive set theory (continuum hypothesis holds for Borel sets, 1916; JFM 46.0291.02), his well known \(G_\delta\)-theorem in topology (each \(G_\delta\)-set in a complete metric space is homeomorphic to a complete metric space, 1924; JFM 50.0141.01), two papers on extensions of continuous mappings (1930; JFM 56.0508.03, 1938; Zbl 0018.27704 and JFM 64.0621.03), improvements of earlier results by Sierpiński, Mazurkiewicz, Kuratowski and some others.

Parts III and IV present some excerpts from Hausdorff’s unpublished legacy. The whole posthumous legacy is too voluminous to publish: it consists of 26 000 pages, of which 1000 are concerned with descriptive set theory and topology. But it is also too interesting to ignore it altogether, and so Part III contains a selection on descriptive set theory divided into six thematic sections: \(\delta s\)-operations (analytic operations); systems of sets, Borel sets, separation; Borel functions; reducible sets; Suslin sets, indices, separation; varia. Excerpts from a particular fascicle are accompanied by editorial remarks and each section is completed by a comprehensive commentary.

Part IV contains a selection on topology. It starts with three fascicles strongly related to Hausdorff’s published papers and so left without comments. These are followed by four remarkable notes on metric spaces, supplied with comments. And they in turn are followed by four very interesting Hausdorff’s essays on basic topological constructions, on curves, arcs and Peano continua, on dimension theory, and on combinatorial topology. The essays are interesting in themselves but now even more as a historical source by offering Hausdorff’s view upon the development of specific areas. The last essay (on combinatorial topology) is preceded by a valuable article by E. Scholz “Hausdorff’s view upon the beginning of algebraic topology”. It should be noted that many Hausdorff notes, now in the legacy, were written during hopelessly darkening period in his life, with no chance for publication; e.g., his proof of the topological invariance of homology groups was completed in June 1940.

Carefully edited, as are the other volumes in this series, this Volume III is an equally important contribution to the history of 20th century mathematics.

Reviewer: Roman Duda (Wrocław)

### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

01A60 | History of mathematics in the 20th century |