Covering theorems

R. L. Moore
1926 Bulletin of the American Mathematical Society  
A set of segments will be said to cover a point set F in the Vitali sense if, for every point P which belongs to F and every positive number e, there exists a segment of G which contains P and is of length less than e. J. Splawa-Neyman has shown f that if, in space of one dimension, F is a closed and bounded point set of measure zero and G is a set of segments which covers F in the Vitali sense, then, for every positive number e, the set G contains a subset G e such that G e covers F and such
more » ... covers F and such that the sum of the lengths of the segments of the set G ê is less than e. He cites the question, raised by Sierpinski, whether this theorem remains true after the removal of the condition that F be closed. I have recently J answered this question in the negative. Splawa-Neyman shows, by an example, that his theorem does not hold true for two dimensions, but makes the following statement, without proof: "Remarquons que notre théorème subsiste pour les espaces à n dimensions, s'il existe pour tout point p de F une sphère appartenant à F de rayon aussi petit que Ton veut et dont le centré est en p." In the present paper I will show that the theorem thus stated, without proof, by Splawa-Neyman remains true on the removal of the condition that F be closed. I will also show that the condition that each point of F be the center of spheres of * Presented to the Society, in a somewhat different form, December 30, 1924. In the abstract of this paper printed in this BULLETIN, vol. 31 (1925), pp. 219-220, proposition (2) is not correctly worded and, as will be shown below, (3) is false. t Sur un théorème métrique concernant les ensembles fermés, FUNDA- MENTA MATHEMATICAE, vol. 5 (1924), pp. 328-330. % Cf. R. L. Moore, Concerning sets of segments which cover a point set in the Vitali sense, PROCEEDINGS OF THE NATIONAL ACADEMY, vol. 10 (1924), pp. 464-467. * These theorems will be stated in the terminology of space of two dimensions. It is easy to see however that this restriction is not necessary, t By a circular region is meant the interior of a circle.
doi:10.1090/s0002-9904-1926-04207-5 fatcat:72qrwmy3m5b7nhkm2jdouiekby