The "normal sets" are separable metric spaces with no isolated points, as introduced by Fréchet in Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo 22, 1-74 (1906).
For an extensive discussion of Brouwer's paper in the historical context see D.M. Johnson's 1981 article in the Archive for History of Exact Sciences.