Mathematics. — ''On linear inner limiting .<iets" ^). By Prof. L. E. J. 

 Brouwer. 



(Gommunicaled in the meeting of April 27, 1917). 



We consider an inner limiting set /, determined inside the unit 

 interval as tiie intersection (greatest common divisor) of the sets of 

 (non-overlapping) intervals i,, i,, . . ., each point of /v+i being also 

 a point of /,. Then the complementary set C of / with regard to 

 the closed nnit interval is the nnion (common measure) of the closed 

 sets ^1, c/,, . . . , each //v+i containing a,. We shall suppose that I 

 as n)ell as C is uncountable in each sub-interval of the unit interval ; 

 then we may assume that each a.^ contains as its nucleus a perfect 

 set p,. The difference of a, and />. will be indicated by v,, the 

 complementary set of />,, considered as a set of intervals, by n, , 

 and the inner limiting set determined as the intersection of ?/i, i<,, .. . 

 by U. Then the points of each u, lie everywhere dense, and each ?/v 

 is a set of order-type i] of intervals, whose length does not exceed 

 a certain value f.. having the limit zero for indefinitely increasing!'. 



Let us assume that we dispose of such a set ;'., of order-type ij of 

 intervals each being an element of one of the seisin, Wv+i, u.j^^i, . . ., 

 that j, contains no point of ?%, but does contain all points of ^ not 

 belonging to v, . We shall indicate a method leading from /.; to such 

 a vset j.,-^i of order-type »j of intervals each lying inside an interval 

 of /v, and being an element of one of the sets Wv-fi, Uy-1^2, u-^-\-:i, • ■ • , 

 that /v-i-i contains no point of v.^i, but does contain all points of /7 

 not belonging to v.^i, each interval of /v containing a subset of 7^-1-1 

 of order-type i]. 



Let AB be an arbitrary element of /. being at the same time an 

 element of ?/,y (ft > v)- let f be the subset of lu+i lying inside AB, 



^) To the last footnote of my former communication on inner limiting sets (these 

 Proceedings XVlll, p. 49) must be added that the changed form in which Schoenflies 

 has referred to my reasoning (applying it to a special case only, and deducing 

 the general theorem from this special case) is irrelevant. The error is contained 

 in the sentence (Entwickelung der Mengenlehre i, p. 359. line 5—8 from the top): 

 "1st namlich P irgend eine abzahlbare Menge, die nicht dicht in bezug auf eine 

 perfekte Menge ist, und geht man durch Hinzufiigung samllicher Grenzpunkte zu 

 einer ahgeschlossenen Menge Q über, so kann diese keinen perfekten Bestandteil 

 enthalten ist also eben falls abzahlbar". 



