The role of the Pompeiu-Hausdorff metric in fixed point theory

The main aim of this note is to highlight the role of the Pompeiu-Hausdorff metric in fixed point theory and, subsidiarily, to touch some issues related to the history of this fundamental concept in modern mathematics. This will allow us to conclude that what is nowadays almost generally called Hausdorff metric (distance) and very seldom Hausdorff-Pompeiu metric (distance) or Pompeiu-Hausdorff metric (distance), should be fairly and correctly named Pompeiu-Hausdorff metric (distance).