 Thus, for instance the familiar Euclidean space has compactness degree zero; it is not compact itself, but every point has a neighborhood bounded by a compact sphere.This compactness degree, zero, equals the dimension of the single point that may be added to Euclidean space to form its one-point compactification.A detailed review of de Groot's compactness degree problem and its relation to other definitions of dimension for topological spaces is provided by Koetsier and van Mill

In 1959, his work on the classification of homeomorphisms led to the theorem that one can find a large cardinal number, ℶ2, of pairwise non-homeomorphic connected subsets of the Euclidean plane, such that none of these sets has any nontrivial continuous function mapping it into itself or any other of these sets.