48 



Mathematics. — "Remark on inner limiting sets". By Prof. L. E. J. 

 Bhouwer. 



(Goinmunicated in the meeting of April 23, 1915). 



The notion of inner limiting set i. e. the set of all the points 

 common to a series of sets of regions, was prepared by Borel '), 

 and fully developed by Young'). The two principal theorems about 

 this class of sets are the following : 



1. An inner limiting set containing a coinijonent dense in itsel/, 

 has the continuous potency. 



2. .1 coimtable set containing no component dense in itself, is an 

 inner limiting set. 



The former theorem has been proved by Young, first for the 

 linear domain, then for the space of 7i dimensions '). The latter 

 theorem has been proved for (he first time by Hobson"*). It is true 

 that this theorem can be considered as a corollary of the following 

 theorem enunciated somewhat before by Young'): 



3. //' Q be an arbitrary set of j>oinis, an inner limiting set exists 

 containing besides Q only limiting points of the ultimate coherence ') 



of Q; 



but this theorem was deduced by Young *) from the property : 

 "Each of the successive adherences ') of a set of points consists 

 entirely of points lohich are limiting points of every p7-eceding 

 adherence', and the proof given by Young for this property is 

 erroneous*), so that undoubtedly the priority for the proof of tlieorem 

 2 belongs to Hobson. 



We can, however, arrive at theorem 2 in a much simpler wa^> 



1) LeQons sur la theorie des fonetions, p. 44. 



2) Leipziger Ber. 1903, p. 288; Proc. London M. S. (2) 3, p. 372. 



3} Leipziger Ber. 1903, p. 289—292; Proc. London M. S. (2) 3, p. 372—374. 

 These proofs are referred to not quite exactly by Schoenflies, Bericht fiber die 

 Mengenlehre II, p. 81 and Entwickelung der Mengenlehre I, p. 356. 



+) Proc. London M. S. (2) 2, p. 316-323. 



6) Proc. London M. S. (2) 1, p. 262—266. 



8) Young, Quarterly Journ of Math., vol. 35, p. 113. 



") Cantor, Acta Mathematica 7, p. 110. 



**) Quarterly Journ. of Math., vol. 35, p. 115. The error is contained in the 

 sentence (line 8—6 from the bottom): "Thus P, being a limiting point of every 

 one of the derived coherences, is a limiting point of f ". A correct proof of 

 the property in question was communicated to me about two years ago by 

 G. Ghisholm Young. 



