This publication represents the PhD thesis of Dimitrie Pompeiu, defended at the Faculté de Sciences de Paris before a committee chaired by Henri Poincaré .
Pompeiu introduced here the distance between two sets, which he called the "écart" and "écart mutuel". The distance between sets is now usually called the "Hausdorff distance" since the idea also appeared in Hausdorff's famous book Grundzuege der Mengenlehre (1914). In this work Hausdorff gives a slightly different definition of distance between sets but he also credits Pompeiu's work and shows that the two definitions give the same topology .
The Pompeiu-Hausdorff distance has been widely used in mathematics.
 Birsan, T., & Tiba, D. (2006). One Hundred Years Since the Introduction of the Set Distance by Dimitrie Pompeiu. In F. Ceragioli, A. Dontchev, H. Futura, K. Marti, & L. Pandolfi (Eds.), System Modeling and Optimization (Vol. 199, p. 35-39). Springer US. doi:10.1007/0-387-33006-2_4
 O'Connor, J. J., & Robertson, E. F. (2007). Dimitrie Pompeiu. MacTutor History of Mathematics.