PRESENT PROBLEMS OF GEOMETRY 567 



is continuous but not one-to-one (Peano, 1890). In the domain of 

 continuous one-to-one correspondence, however, spaces of different 

 dimensions are not equivalent (Jiirgens, 1899). 



An important class of curves, much more general than those 

 referred to above, consists of those point sets which are equivalent 

 (in the sense of analysis situs) to the straight line or segment of a 

 straight line. This is Hurwitz's simple and elegant geometric form- 

 ulation of the concept originally treated analytically by Jordan, 

 the most fundamental curve concept of to-day. The closed Jordan 

 curves are defined in analogous fashion as equivalent to the peri- 

 meter of a square (or the circumference of a circle). 



A curve of this kind divides the remaining points of the plane into 

 two simply connected continua, an inside and an outside. The 

 necessity for proof of this seemingly obvious result is seen from the 

 fact that the Jordan class includes such extraordinary types as the 

 curve with positive content constructed recently by Osgood. 1 Such 

 a separation of the plane may, however, be thought about by other 

 than Jordan curves: the concept of the boundary of a connected 

 region gives perhaps the most extensive class of point sets which 

 deserve to be called curve. Schoenflies proposes a definition for the 

 idea of a simple closed curve which makes it appear as the natural 

 extension, in a certain sense, of the polygon: a perfect set of points 

 P which separates the plane into an exterior region E and an interior 

 region / such that any E point can be connected with any / point 

 by a path (Polygonstrecke) having only one point in common with 

 P. This is in effect a converse of Jordan's theorem, and shows 

 precisely how the Jordan curve is distinguished from other types 

 of boundaries of connected regions. 



These discussions are mentioned here simply as aspects of a really 

 fundamental problem: the revision of the concepts and results of 

 that division of geometry which has been variously termed analysis 

 situs, theory of connection, topology, geometry of situation a 

 revision to be carried out in the light of the theory of assemblages. 2 



Algebraic Surfaces and Birational Transformations 



After the demonstration of the power of the methods based upon 

 projective transformation, the chief contribution due to the 

 geometers of the first half of the nineteenth century, attempts 

 were made to introduce other types of one-to-one correspondence or 

 transformation into algebraic geometry; in particular the inversion 

 of William Thomson and Liouville, and the quadratic transformation 

 of Magnus. The general theory of such Cremona transformations 

 was inaugurated by the Italian geometer in his memoir Sulle tras- 



1 Trans. Amer. Math. Soc., vol. iv (1903), p. 107. 



2 Cf. Schoenflies, Math. Annalen, vols. LVIII, LIX (1903, 1904). 



