Johannes de Groot - Research

Research

De Groot published approximately 90 scientific papers. His mathematical research concerned, in general, topology and topological group theory, although he also made contributions to abstract algebra and mathematical analysis.

He wrote several papers on dimension theory (a topic that had also been of interest to Brouwer). His first work on this subject, in his thesis, concerned the compactness degree of a space: this is a number, defined to be −1 for a compact space, and 1 + x if every point in the space has a neighbourhood the boundary of which has compactness degree x. He made an important conjecture, only solved much later in 1982 by Pol and 1988 by Kimura, that the compactness degree was the same as the minimum dimension of a set that could be adjoined to the space to compactify it. 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. 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 topologically complete spaces in terms of cocompactness, and a topological characterization of Hilbert space. From 1962 onwards, his research primarily concerned the development of new topological theories: subcompactness, cocompactness, cotopology, GA-compactification, superextension, minusspaces, antispaces, and squarecompactness.