##
**Collected works. Vol. II: Foundations of set theory.
(Gesammelte Werke. Band II: Grundzüge der Mengenlehre. Herausgegeben von E. Brieskorn, S. D. Chatterji, M. Epple, U. Felgner, H. Herrlich, M. Hušek, V. Kanovei, P. Koepke, G. Preuß, W. Purkert und E. Scholz.)**
*(German)*
Zbl 1010.01031

Berlin: Springer. xviii, 883 S. EUR 99.95 (D); sFr. 155.00 (2002).

For a review of Vol. IV see Zbl 0980.01026.

This volume of Felix Hausdorff’s (1868-1942) Werke contains a reprint of his most influential work, Grundzüge der Mengenlehre from 1914 (see JFM 45.0123.01), together with many additions to that. The volume starts with the list of Hausdorff’s publications (reprinted in each volume of Werke), followed by a historical introduction to Grundzüge, the reprint of Grundzüge, remarks to Grundzüge, essays on particular topics influenced by Grundzüge, some posthumous works of Hausdorff, his self-review and 8 reviews of Grundzüge from the years 1914-1922, index of names, subject index. Altogether 883 pages, of which 486 belong to Grundzüge itself and the rest to additions.

An extensive historical introduction by W. Purkert consists of three parts: Hausdorff’s way to Grundzüge, a survey of monographs and textbooks on the theory of sets up to 1914 (when Grundzüge appeared), the reception of Grundzüge (a section in it, that on propagation of set-theoretic methods in topology, was written by E. Brieskorn and L. Scholz). The text of Grundzüge is a photocopy of the original one, with a few added numbers, indicating remarks, on margins only. The remarks (totalling 102) are well placed short historical comments on particular Hausdorff’s statements; each comment has its author, one from among: U. Felgner, H. Herrlich, V. Kanovei, P. Koepke, G. Preuß, and W. Purkert.

The topics covered by essays are eleven: the concept of a function (the author of this one and the next two is U. Felgner), the concept of a cardinal number, Hausdorff theory of some particular sets and its influence, the concept of a topological space (a very interesting essay, 70 pages long, by five authors: M. Epple, H. Herrlich, M. Hušek, G. Preuß, W. Purkert, E. Scholz), separation axioms (this one and the next four are authored jointly by H. Herrlich, M. Hušek, and G. Preuß), connectedness, axioms of countability, Hausdorff metric and hyperspaces, completeness and total boundedness, descriptive set theory (V. Kanovei, P. Koepke), measure and integration theory (S. D. Chatterji).

Chosen posthumous works are concerned with particular topological topics like neighbourhood axioms, open mappings, other types of axioms, separability etc.; some are provided with commentaries.

Then there is a Hausdorff’s self-review for Jahresbericht and eight other reviews, of which particularly interesting are two most extensive ones by G. Vivanti and by H. Blumberg. A short and anonymous French review from 1916 shows that Grundzüge exerted an influence even over fronts of raging War World I.

In short, the mathematical community has received a magnificent and important volume, a result of assiduous work of the whole team. It is not only a tribute to the talent and achievements of a great man, but it should also be indispensable for any historian of modern mathematics and in any mathematical library.

This volume of Felix Hausdorff’s (1868-1942) Werke contains a reprint of his most influential work, Grundzüge der Mengenlehre from 1914 (see JFM 45.0123.01), together with many additions to that. The volume starts with the list of Hausdorff’s publications (reprinted in each volume of Werke), followed by a historical introduction to Grundzüge, the reprint of Grundzüge, remarks to Grundzüge, essays on particular topics influenced by Grundzüge, some posthumous works of Hausdorff, his self-review and 8 reviews of Grundzüge from the years 1914-1922, index of names, subject index. Altogether 883 pages, of which 486 belong to Grundzüge itself and the rest to additions.

An extensive historical introduction by W. Purkert consists of three parts: Hausdorff’s way to Grundzüge, a survey of monographs and textbooks on the theory of sets up to 1914 (when Grundzüge appeared), the reception of Grundzüge (a section in it, that on propagation of set-theoretic methods in topology, was written by E. Brieskorn and L. Scholz). The text of Grundzüge is a photocopy of the original one, with a few added numbers, indicating remarks, on margins only. The remarks (totalling 102) are well placed short historical comments on particular Hausdorff’s statements; each comment has its author, one from among: U. Felgner, H. Herrlich, V. Kanovei, P. Koepke, G. Preuß, and W. Purkert.

The topics covered by essays are eleven: the concept of a function (the author of this one and the next two is U. Felgner), the concept of a cardinal number, Hausdorff theory of some particular sets and its influence, the concept of a topological space (a very interesting essay, 70 pages long, by five authors: M. Epple, H. Herrlich, M. Hušek, G. Preuß, W. Purkert, E. Scholz), separation axioms (this one and the next four are authored jointly by H. Herrlich, M. Hušek, and G. Preuß), connectedness, axioms of countability, Hausdorff metric and hyperspaces, completeness and total boundedness, descriptive set theory (V. Kanovei, P. Koepke), measure and integration theory (S. D. Chatterji).

Chosen posthumous works are concerned with particular topological topics like neighbourhood axioms, open mappings, other types of axioms, separability etc.; some are provided with commentaries.

Then there is a Hausdorff’s self-review for Jahresbericht and eight other reviews, of which particularly interesting are two most extensive ones by G. Vivanti and by H. Blumberg. A short and anonymous French review from 1916 shows that Grundzüge exerted an influence even over fronts of raging War World I.

In short, the mathematical community has received a magnificent and important volume, a result of assiduous work of the whole team. It is not only a tribute to the talent and achievements of a great man, but it should also be indispensable for any historian of modern mathematics and in any mathematical library.

Reviewer: Roman Duda (Wrocław)

### MSC:

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

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