Ralph Kopperman

Computer graphics and connected topologies on finite ordered sets

Abstract Motivated by a problem in computer graphics, we develop a finite analog of the Jordan curve theorem in the following context. We define a connected topology on a finite ordered set; our plane is then a product of two such spaces with the product topology.

Asymmetry and duality in topology

Abstract Many mathematical structures come in symmetric and asymmetric versions. Classical examples include commutative and noncommutative algebraic structures, as well as symmetric preorders (=equivalence relations) and asymmetric such (usually partial orders). In these cases, there is always a duality available, whose use simplifies their study, and which reduces to the identity in the symmetric case. Also, in each of these cases, while symmetry is a simplifying assumption, there are many useful asymmetric examples. A similar phenomenon occurs in general topology, although in this case there are often many available useful duals. There are also many useful asymmetric spaces, such as the finite T0 spaces and the unit interval with the upper, or lower topology (in fact the Scott and lower topologies on any continuous lattice). The latter, using a dual, gives rise to the usual topology and order on the unit interval.

All topologies come from generalized metrics

RALPH KOPPERMAN: I received my Ph.D. from M.I.T. in 1965, and came to The City College in 1967. SEEK is a special program for students from poor areas of the city, and I have coordinated its mathematics efforts since 1969. My research interest has always been in limits, which I first tried to study through infinitary languages (logic). I was able to publish in the field, but was not happy with the results of that research, and went into point-set topology in 1980. I was a founding member of the CCNY Seminar on General Topology and Topological Algebra in 1981, and have been involved with it ever since.

Boundaries in Digital Planes

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.

Continuity spaces: reconciling domains and metric spaces

Abstract We use continuity spaces, a common refinement of posets and metric spaces, to develop a general theory of semantic domains which includes metric spaces and domains of cpos as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming. Our proposal for a general notion of semantic domain is a symmetrically compact -continuity space, where is a value quantale. We show that the category of symmetrically compact -continuity spaces with continuous maps has many of the key properties required of a category of domains and that it captures, in a natural way, the traditional examples. In general, the category will not be Cartesian closed; however, powerdomains do exist and, by adapting a construction of Suenderhauf to continuity spaces, we show that they define a computational monad in the sense of Moggi.

Dimensional properties of graphs and digital spaces

Many applications of digital image processing now deal with three- and higher-dimensional images. One way to represent n-dimensional digital images is to use the specialization graphs of subspaces of the Alexandroff topological space ℤn (where ℤ denotes the integers with the Khalimsky line topology). In this paper the dimension of any such graph is defined in three ways, and the equivalence of the three definitions is established. Two of the definitions have a geometric basis and are closely related to the topological definition of inductive dimension; the third extends the Alexandroff dimension to graphs. Diagrams are given of graphs that are dimensionally correct discrete models of Euclidean spaces, n-dimensional spheres, a projective plane and a torus. New characterizations of n-dimensional (digital) surfaces are presented. Finally, the local structure of the space ℤn is analyzed, and it is shown that ℤn is an n-dimensional surface for all n≥1.

Some Topology-based Image Processing Algorithms

ABSTRACT: Some results on certain finite and locally finite spaces which may be used in image processing are reviewed, and algorithms related to these results are given and discussed.

Computational Models for Ultrametric Spaces

Abstract For every ultrametric space, the set of closed balls of radius 0 or 2 - n for some n , form an algebraic poset under reverse inclusion. If the ultrametric space is complete separable, then they form a Scott computational model for it. Conversely, every topological space with an algebraic computational model is a complete separable ultrametrizable space.

The Khalimsky Line as a Foundation for Digital Topology

An object is defined from which digital spaces can be built. It combines the “one-dimensional connectedness” of intervals of reals with a“point-bypoint” quality necessary for constructing algorithms, and thus serves as a foundation for digital topology. Ideas expressed in quotation marks here are given precise meanings. This study considers the Khalimsky line, that is, the integers, equipped with the topology in which a set is open iff whenever it contains an even integer, it also contains its adjacent integers. It is shown that this space and its interval subspaces are those satisfying the conditions mentioned previously. The Khalimsky line is used to study digital connectedness and homotopy.

Lengths on semigroups and groups

We give a characterization of those semigroups with topology arising from a collection of left sub-invariant pseudometrics or quasimetrics. We also characterize those with topology arising from sub-invariant pseudometrics or quasimetrics.