An uncountable, closed, and non-dense point set each of whose complementary intervals abuts on another one at each of its ends
Bulletin of the American Mathematical Society
On page 92 of the 1907 edition of Hobson's The Theory of Functions of a Real Variable, and again on page 113 of the second edition of the same treatise, there occurs the following statement: f "A non-dense closed set is enumerable if its complementary intervals are such that every one of them abuts on another one at each of its ends" To prove this statement, Hobson lets G denote the nondense closed set in question and argues, in part, as follows: "In this case, all the points of G are either
... of G are either end-points of adjacent intervals, or limiting points, on both sides, of a sequence of such end-points; unless a J or b $ be a limiting point, in which case it belongs to G. The end-points have the same cardinal number as the rational numbers, since the set of intervals is enumerable. Moreover the external § points form a finite set, or an enumerable set; because to each such external point there corresponds an enumerable set of endpoints of which it is the limiting point and in this correspondence any one end-point can correspond to at most two limiting points, one on each side of it. 1 ' Just what is the meaning of the above italicized statement? If P is an external point, then, by definition, there exists a * Presented to the Society, December 29, 1922. t This statement will be called Proposition A. J Here a and b apparently denote the end-points of some interval which contains the set G. § External points are defined by Hobson as points such as are not end-points of any contiguous interval but are limit points on both sides of such end-points.