Topologie I. By P. Alexandroff and H. Hopf. Pp. xiv, 636. RM. 46.80. 1935. Die Grundlehren der Mathematischen Wissenschaften, 45. (Springer)
1937
Mathematical Gazette
296 ÏHE MATHEMATICAL GAZETTE EEV1EWS. Topologie I. By P. ALEXANDROFF and H. HOPF. Pp. xiv, 636. RM. 46.80. 1935. Die Grundlehren der Mathematischen Wissenschaften, 45. (Springer) About the year 1925, topology was in the uncomfortable condition of falling into two scarcely connected parts, which had for some time threatened to become completely separate subjects. These were the theory of sets of points in generál topological spaces-" point-set topology "-and the algebraic theory of the
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... ty properties of manifolds and complexes-" combinatorial topology ". These two subjects had distinct origins, the first in the work of Cantor, Borel, and others, the second in Poincaré's famous series of memoirs, starting in 1895, on the Analysis Situs of n-dimensional manifolds. In the beginning the only connection between the two branches was that the object of both was to discover " topological " properties-properties that are invariant under a (1,1), both ways continuous, mapping. Poincaré's rather intuitive treatment was soon felt to be logically unsatisfactory, and it was clear that to provide the combinatorial theory with a rigorous basis some appeal must be made to point-set theorems and methods. This difficult problém was solved, in the astonishingly simple way that is now familiar, by the work of Brouwer and Alexander. A further step was the Duality theorem of Alexan der, which showed that Jordan's theorem, which originally seemed to be on the point-set side of the gap, really belonged to the combinatorial side. So far, however, the approach had been entirely from one side : point-set theor ems of a comparatively simple kind were ušed in the Theory of Complexes, but the point-set theory itself pursued a course quite untouched by combinatorial methods. This situation was completely changed by papers that began to appear about 1925, by Vietoris, Lefschetz, but above all by Alexandroff, in which, on the one hand, the problém of extending the concepts and results of the Theory of Complexes to generál topological (or at least generál metrical) spaces was systematically undertaken; on the other hand, theories that had been regarded hitherto as definitely in point-set territory (in particular the Theory of Dimension) were fruitfully attacked by combinatorial methods. An important part in this development was played by Alexandroff's method of approximating to metrical spaces by an abstract complex called the nerve, determined essentially by the way in which a systém of closed sets, covering the space and of small diameter, fitted together. Lebesgue's famous " Pnastersatz " was the basis of this method. By the time this unification began both sides of Topology had attained a high pitch of development. It is therefore not surprising that the combinatorial theory of generál spaces, which takés both sides of the subject, at more or less their 1925 levels, as starting point, should be beyond the scope of any ordinary book for students, and in fact (though Lefschetz's Topology (1930) has a chapter on open manifolds) no textbook has hitherto attempted any systematic account of the blended theory. (The excellent Encyclopedia article of Tietze and Vietoris has of course no pretensions to be a textbook. It is an account of the statě of knowledge in 1931, with references but not proofs.) Nevertheless to stop short at this stage is to give a most inadequate idea of modern topology as a unity. Alexandroff and Hopf háve therefore set about repairing this omission by writing a treatise on such a scale that they will be able to give an account of the main lineš of the entire theory, an account, as they say in their Preface, " not of the whole of topology, but of topology as a whole." Their framework is a work in three volumes, of which the first, now published, https://doi.
doi:10.1017/s0025557200058137
fatcat:co45l3lvpjepxjbid5ualue5ce