 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.The topological spaces formed by these subsets of the plane thus have a trivial automorphism group; de Groot used this construction to show that all groups are the automorphism group of some compact Hausdorff space, by replacing the edges of a Cayley graph of the group by spaces with no nontrivial automorphisms and then applying the Stone–Čech compactification.A related algebraic result is that every group is the automorphism group of a commutative ring.