Mathematics. — "Inner Limiting Sets". By Prof. J. Wolff. 

 (Coiiimiinicated by Prof, Hkndrik dk Vries). 



(Communicated at the meeting of February 24, 1923). 



HoBSON lias been the first to prove tiie following- theorem:') 



An emtmerabh set of points which has no part that is dense in 

 itself, is an inner limiting set, i.e. the common part of an enumerable 

 set of open sets each of which we may assume to contain the 

 following one. 



Brouwer has given an extremel_y short proof, imt just as Hobson 

 he makes use of the transfinile ordinal numbers''). 



In the proof which follows here, no use is made of these numbers. 



1. If Ej, E,, . . . are inner limiting sets, if further each Ejc is 

 a part of an open set i2k, while no two i^k liave any points in 

 common, also the sum E^ -\- E, ~\- . . . is an inner limiting set. . 



For we may write: 



Ek = £iki^k2 k = 1.2 



which means that Eic is the set of points lying in 42^, for every i- 

 The i2ki are open sets of which we may assume that they all lie 

 in S*k- The set 



(.Q„+.o, + ...)(<2„ + i2,. + ...).... 



contains E^ -\- E^ -\- but no point outside them, as iiui^-ij = '^ 



for kjLl. Now the auxiliary theorem has been proved. 



'1. We call a set of points E an inner limiting set in a pohil P 

 if there exists an open set containing this point, so that the part 

 of E lying in this set is an inner limiting set. This holds also 

 good for the part of E lying in an arbitrary open set which is a 

 part of the above mentioned one. 



3. If an enumerable set E is an inner limiting set in each of 

 its points, E is an inner limiting set. 

 We call the points of E: P„ P„ . . . 



') Proc. London M.S. (2) 2, p. 316—323. 



') These Proceedings, Vol. XVIll p. 48 (1915). * 



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