 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.Other results in his research include a proof that a metrizable topological space has a non-Archimedean metric (satisfying the "strong triangle inequality" "d"("x","z") ≤ max("d"("x","y"),"d"("y","z")) if and only if it has dimension zero, description of completely metrizable spaces in terms of cocompactness, and a topological characterization of Hilbert space.