New York Topology Seminar

• Ralph Kopperman

External page: New York Topology Seminar

All talks

• Thursday, November 01, 2007, 04:00PM, Mathematics Dept., Baruch Coll. Vertical Campus,

  Venu Menon (Univ. of Connecticut), Continuity in partially ordered sets

  Continuous lattices and their generalizations, continuous domains, have been studied for more than three decades. Recall that given elements of a poset, p is way below q, if whenever q is less than or equal to each upper bound of a directed set D, then p is less than or equal to an element of D. Continuous lattices are complete lattices where each element is the supremum of elements way below it. A poset is a continuous domain, if it has sups of directed sets and satisfies the following two conditions: (i) each element is the sup of elements way below it, and (ii) for each element, the set of elements way below it is directed.

  A continuous poset is any poset in which the conditions (i) and (ii) are satisfied. In a complete lattice, in fact in any sup-semilattice, condition (ii) above is automatically satisfied. The purpose of this talk is to look at posets which need not be dcpos or lattices but which satisfy the condition that each element is the sup of elements way below it. Since we don't require the set of elements way below any element to be directed, we will require a condition slightly stronger than condition (i) above. Several of the pleasing algebraic and topological properties of continuous domains extend to this setting.

• Thursday, October 18, 2007, 04:00PM, CCNY, NAC 4/205

  Homeira Pajoohesh (Medgar Evers College, CUNY), Generalized metric topologies on lattice ordered groups

  "Intrinsic" metrics into lattice ordered groups have long been considered. Topologies on such groups have been considered as well. But topologies often arise from metrics into the reals, and getting topologies from these metrics into lattice ordered groups has not been discussed. A key issue must be overcome: properties of the strictly positive reals key to defining the metric topology fail for the strictly positive elements of a typical lattice ordered group. We solve this problem, and show how some important topologies indeed arise from metrics.

  We define partial metrics and discuss their uses in computer science, in obtaining intrinsic topologies on lattice ordered groups, and elsewhere. A central use for them is to split certain metric topologies on ordered sets, into upper and lower subtopologies. Among these is the Euclidean topology on the lattice ordered group of real numbers.

• Thursday, October 04, 2007, 04:00PM, Long Island University, CW Post Center

  Ralph Kopperman (City College of CUNY), Normal non-Hausdorff spaces

  Beginning with the Sierpinski space - the smallest non-trivial space - examples of such spaces abound. Far from simply being curiosities, they are needed to approximate compact Hausdroff spaces with finite T0 spaces.

  We give examples and a partial characterization of normal non-Hausdorff spaces. We discuss the motivation for this type of approximation in computer storage, and the role of these spaces in it.

  Tea 3:15, Pell Hall 240; talk at 4:00 Pell Hall, room to be arranged. For more information and parking, write to SAndima@liu.edu, or call the C.W. Post Math Department at 516-299-2448.