( 138 ) 



Theorem 1 . ^1 ivell-ordei-ed set of points in Sj),. edck point of inhich 

 possesses a jiiilte distance from the set formed htj aU the following 

 points, is denumerable. 



Out of this is obtained in the following form a generalizalion of 

 Cantor's fundamental theorem, probably the widest one of which it 

 is ea[)able : 



When a closed set of points is replaced b}- a closed set contained 

 in it, we shall say that the first set is lopped. 



A fundamental series of closed sets of ])oints will be called a 

 lopping series, if each following set is contained in the preceding 

 one. The greatest common part of the terms of such a series is a 

 closed set, which we shall call the limifint/ set of ihe lopiung series. 



By an Inductihle pi'opei-tij of closed sets of points we shall under- 

 stand a [)ro{)erly which, when possessed by each term of a l()p[)ing 

 series, holds also for the limiting set of that series. 



From theorem 1 now follows : 



Theorem 2. Let n be ri closed set of points of Sp^ possessing the 

 inductihle property a; ive can reduce it by a denumerable number of 

 lojtpings of a defnite kind j? to a closed set of points n, possessing 

 stiU the property c, but losing it by any new lopping of kind ,?. 



This theorem can be specialized in many directions. 



If we choose as property n the simple [)roperty of being closed, 

 and as lopping of kind ,i the destruction of an isolated j)oint resp. 

 of an isolated piece, then Cantor's fundamental theorem resp. its 

 Schoen flies extension appears. 



An other special case is obtained in the following way : 



After ZoRETTi ^' a continuum C is called Irreductible between Pand 

 Q, if the pair of points {1\Q) belongs to C, but to no other conti- 

 nuum contained in C, and Janiszewski-) and Mazurkiewicz^) have 

 proved the following theorem : 



Let C be an arbitrary continuum and P and Q two of its points, 

 then in C is contained a continuum irreductible betweeii P and Q. 



This property appears likewise as a special case of theorem 2, 

 namely by choosing as property « the property of containing Pand 

 Q and being continuous, and as lopping of kind {i the most general 

 lopping. 



§ 2. 

 The structure of closed sets of pieces. 



In § 3 of the first communication it has been proved that all perfect 



1) Annales de TÉcole Normale, 1909, p. 485. 

 -) Goraptes Rendus, t. 15i, p. 198. 

 3) ibid., p. 296. 



