Olaf Klinke

Papers and slides

  • A bitopological point-free approach to compactifications PDF
    Final version of my PhD thesis.
  • On the soberification of the weak lower topology PS
    Joint work with M.A. Moshier. The weak lower topology on a partially ordered set has as basic closed sets the finitely generated upper sets. We show that the weak lower topology is isomorphic to the MacNeille completion of the free meet-semilattice over the poset. Using this characterisation we study the sober space arising from a poset in its weak lower topology as well as the spectrum of the weak lower topology regarded as a bounded distributive lattice.
  • A bitopological approach to compactification PDF PS
    Paper published in Topology and Applications
  • Point-free compactifications of bitopological spaces PDF
    Presentation for the Domains X conference in Swansea September 2011.
  • Slides: Pointfree bitopological compactification PDF
    Slides from my talk at the 3rd Annual Workshop in Pointfree Topology at Chapman university.
  • Slides: A relational representation of domains PDF
    A presentation prepared for the BCTCS conference 2011. Starting from the concept of Vickers' information systems for presenting continuous dcpos, we define a similar category which can accommodate richer algebraic structures such as preframes and continuous lattices in a clear and systematic way. As with information systems, some duality theorems from domain theory become almost trivial to prove, as they are built in to our structures. See also Witness sets for the continuity of posets below.
  • Slides: The Swiss Army knife of (pointfree) compactifications PDF
    Slides from my talk at the Summer Topology conference 2010 in Kielce. We bring together ideas from bitopology and M. Smyth's stable compactification. Smyth showed that, using a suitable auxiliary relation on the open set lattice of a space, one obtains the topology of a stable compactification via the lattice of round ideals of opens. Here we break down the auxiliary relation into a composite of two more fundamental relations between pairs of lattices. One of these lattices is thought of as the opens, while the other lattice symbolises the opens of the cocompact topology. Since every Hausdorff compactification arises from a suitable strong inclusion on the open set lattice, our compactification subsumes all classical compactifications.
  • Slides: What topology does for computer science - Postgraduate seminar PDF

    I will try to convince the audience that topology, which is a field of mathematics arising from geometry, is useful in computer science when one does computing with infinite data structures such as streams of numbers, infinite trees and alike.

    In particular we discuss different methods of representing real numbers and relate them to geometric ideas. We show how geometric ideas translate quite naturally to issues about deciding properties of numbers such as 'is the number positive?' or 'is the number equal to 0?' or 'can this function be implemented in some programming language?'

  • Sublocales of regular reasonable d-frames PDF PS
    Work in progress. A reasonable d-frame consists of two frames together with two relations between them, which satisfy a number of reasonable axioms. For example, any pair of topologies of a bitopological space gives rise to a reasonable d-frame. Many separation axioms of classical topology, like regularity, have a canonical counterpart in the theory of d-frames. I show that being reasonable is equivalent to the existence of certain Galois connections between the frames and their filter spaces. Just as a locale has sublocales which are given by quotient frames, we investigate which quotients of reasonable d-frames again enjoy the reasonableness axioms and give a concrete description of open sublocales of regular reasonable d-frames.
  • Thesis Proposal (Research Student Monitoring Group Report 3) PDF PS
    The goal of this thesis is to develop a notion of local compactness and related properties for bitopological spaces. This notion has to fit into, and extend, the existing results about stably compact spaces and the Stone duality for bitopological spaces. extended abstract (html)
  • On the 90-degree-lemma PDF PS
    In their technical report On the bitopological nature of Stone duality Jung and Moshier axiomatise a bitopological space as a d-frame, which can equivalently be described as a partial frame, a structure with two orders, one being a special Scott domain and the other a complete lattice. The rich interaction of these two orders arises from a ternary operation on distributive lattices and is informally known as the 90-degree-lemma. Motivation for considering a second order originates in Belnap's four-valued logic. The infinitary connections of the two orders are based on a set of axioms which are derived from the Stone duality for bitopological spaces. In this paper it is shown that the axioms given by Jung and Moshier contain some previously unknown redundancies. The redundancies yield an isomorphism of two categories, one having special Scott domains as objects and the other a certain type of complete lattice.