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Kyriakos Papadopoulos

Publications and source records attributed to Kyriakos Papadopoulos.

33 records · Page 2Linked to original sources
arXiv2018

On Two Topologies that were suggested by Zeeman

The class of Zeeman topologies on spacetimes in the frame of relativity theory is considered to be of powerful intuitive justification, satisfying a sequence of properties with physical meaning, such as the group of homeomorphisms under such a topology is isomorphic to the Lorentz group and dilatations, in Minkowski spacetime, and to the group of homothetic symmetries in any curved spacetime. In this article we focus on two distinct topologies that were suggested by Zeeman as alternatives to his Fine topology, showing their connection with two orders: a timelike and a (non-causal) spacelike one. For the (non-causal) spacelike order, we introduce a partition of the null cone which gives the desired topology invariantly from the choice of the hyperplane of partition. In particular, we observe that these two orders induce topologies within the class of Zeeman topologies, while the two suggested topologies by Zeeman himself are intersection topologies of these two order topologies (respectively) with the manifold topology. We end up with a list of open questions and a discussion, comparing the topologies with bounded against those with unbounded open sets and their possible physical interpretation.

math-ph↗
arXiv2018

Spacetime Singularities vs. Topologies of Zeeman-Göbel Class

In this article we first observe that the Path topology of Hawking, King and MacCarthy is an analogue, in curved spacetimes, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski spacetime. We then review a result of a recent paper on spaces of paths and the Path topology, and see that there are at least five more topologies in the class $\mathfrak{Z}-\mathfrak{G}$ of Zeeman-Göbel topologies which admit a countable basis, incorporate the causal and conformal structures, but the Limit Curve Theorem fails to hold. The "problem" that L.C.T. does not hold can be resolved by "adding back" the light-cones in the basic-open sets of these topologies, and create new basic open sets for new topologies. But, the main question is: do we really need the L.C.T. to hold, and why? Why is the manifold topology, under which the group of homeomorphisms of a spacetime is vast and of no physical significance (Zeeman), more preferable from an appropriate topology in the class $\mathfrak{Z}-\mathfrak{G}$ under which a homeomorphism is an isometry (Göbel)? Since topological conditions that come as a result of a causality requirement are key in the existence of singularities in general relativity, the global topological conditions that one will supply the spacetime manifold might play an important role in describing the transition from the quantum non-local theory to a classical local theory.

math-ph↗
arXiv2018

On the Orderability Problem and the Interval Topology

The class of LOTS (linearly ordered topological spaces, i.e. spaces equipped with a topology generated by a linear order) contains many important spaces, like the set of real numbers, the set of rational numbers and the ordinals. Such spaces have rich topological properties, which are not necessarily hereditary. The Orderability Problem, a very important question on whether a topological space admits a linear order which generates a topology equal to the topology of the space, was given a general solution by J. van Dalen and E. Wattel, in 1973. In this article we first investigate the role of the interval topology in van Dalen's and Wattel's characterization of LOTS, and we then examine ways to extend this model to transitive relations that are not necessarily linear orders.

math.GN↗
arXiv2018

Nests, and their Role in the Orderability Problem

This chapter is divided into two parts. The first part is a survey of some recent results on nests and the orderability problem. The second part consists of results, partial results and open questions, all viewed in the light of nests. From connected LOTS, to products of LOTS and function spaces, up to an order relation in the Fermat Real Line.

math.GN↗
arXiv2018

The Order on the Light Cone and its induced Topology

In this article we first correct a recent misconception about a topology that was suggested by Zeeman as a possible alternative to his Fine topology. This misconception appeared while trying to establish the causality in the ambient boundary-ambient space cosmological model. We then show that this topology is actually the intersection topology (in the sense of G.M. Reed) between the Euclidean topology on $\mathbb{R}^4$ and the order topology whose order, namely horismos, is defined on the light cone. Last, but not least, we show that the order topology from horismos belongs to the class of Zeeman topologies. These results accelerate the need for a deeper and more systematic study of the global topological properties of spacetime manifolds.

math-ph↗
arXiv2018

Robots That Do Not Avoid Obstacles

The motion planning problem is a fundamental problem in robotics, so that every autonomous robot should be able to deal with it. A number of solutions have been proposed and a probabilistic one seems to be quite reasonable. However, here we propose a more adoptive solution that uses fuzzy set theory and we expose this solution next to a sort survey on the recent theory of soft robots, for a future qualitative comparison between the two.

cs.RO↗
arXiv2018

A study concerning splitting and jointly continuous topologies on $C(Y,Z)$

Let $Y$ and $Z$ be two fixed topological spaces and $C(Y,Z)$ the set of all continuous maps from $Y$ into $Z$. We construct and study topologies on $C(Y,Z)$ that we call ${\cal F}_n(τ_n)$-family-open topologies. Furthermore, we find necessary and sufficient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.

math.GN↗
arXiv2017

On Properties of Nests: Some Answers and Questions

By considering nests on a given space, we explore order-theoretical and topological properties that are closely related to the structure of a nest. In particular, we see how subbases given by two dual nests can be an indicator of how close or far are the properties of the space from the structure of a linearly ordered space. Having in mind that the term interlocking nest is a key tool to a general solution of the orderability problem, we give a characterization of interlocking nest via closed sets in the Alexandroff topology and via lower sets, respectively. We also characterize bounded subsets of a given set in terms of nests and, finally, we explore the possibility of characterizing topological groups via properties of nests. All sections are followed by a number of open questions, which may give new directions to the orderability problem.

math.GN↗
arXiv2017

Admissible topologies on $C(Y,Z)$ and ${\cal O}_Z(Y)$

Let $Y$ and $Z$ be two given topological spaces, ${\cal O}(Y)$ (respectively, ${\cal O}(Z)$) the set of all open subsets of $Y$ (respectively, $Z$), and $C(Y,Z)$ the set of all continuous maps from $Y$ to $Z$. We study Scott type topologies on ${\mathcal O}(Y)$ and we construct admissible topologies on $C(Y,Z)$ and ${\mathcal O}_Z(Y)=\{f^{-1}(U)\in {\mathcal O}(Y): f\in C(Y,Z)\ {\rm and}\ U\in {\mathcal O}(Z)\}$, introducing new problems in the field.

math.GN↗
arXiv2017

On the Possibility of Singularities on the Ambient Boundary

The order horismos induces the Zeeman $Z$ topology, which is coarser than the Fine Zeeman Topology $F$. The causal curves in a spacetime under $Z$ are piecewise null. $F$ is considered to be the most physical topology in a spacetime manifold $M$, as the group of homeomorphisms of $M$ is isomorphic to the group of homothetic transformations of $M$. $Z$ was used in the Ambient Boundary-Ambient Space cosmological model, in order to show that there is no possibility of formation of spacetime singularities. In this article we question this result, by reviewing the corresponding articles, and we propose new questions towards the improvement of this model.

gr-qc↗
arXiv2016

The causal order on the ambient boundary

We analyse the causal structure of the ambient boundary, the conformal infinity of the ambient (Poincaré) metric. Using topological tools we show that the only causal relation compatible with the global topology of the boundary spacetime is the horismos order. This has important consequences for the notion of time in the conformal geometry of the ambient boundary.

hep-th↗
arXiv2013

Research talk: On the Classification of Generalized Cartan Matrices of Rank 3

In this talk we describe very briefly how Gritsenko and Nikulin classified in \cite{3} the generalised Cartan matrices of rank 3, of elliptic type (so they have the generalised lattice Weyl vector), which are twisted to symmetric generalised Cartan matrices. We then introduce a conjecture on this classification and propose an algorithm for attacking it. The author would like to thank prof. V.V. Nikulin for giving him the opportunity to study this subject.

math.GR↗