On two topologies that were suggested by Zeeman

Papadopoulos, Kyriakos; Papadopoulos, Basil K.

doi:10.1002/mma.5238

Cited by 5 publications

(9 citation statements)

References 12 publications

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“…Furthermore, we observe that the Limit Curve Theorem holds for each of the topologies 2, 3,8,9,14,15,23,24 of our list, but not for the topologies 5, 6, 11, 12, 17, 18, 20, 21, 26, 27, 29, 30. Following the argument of Low ([5], paragraph V), we can easily see if U is a basic-open set of either of the topologies 2, 3, 8, 9, 14, 15, 23 or 24, then this set does not contain the light cone of the event which defines it. Consider a sequence of null vectors p n converging to p in the usual topology.…”

Section: Singularitiesmentioning

confidence: 69%

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On the Causal and Topological Structure of the 2-Dimensional Minkowski Space

2019

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Section: Singularitiesmentioning

confidence: 69%

“…In [15] (paragraph 1.4), we intuitively (i.e. in a topological sense, invariantly from a change in the geometry) partitioned the light-cone so that apart from future and past we also achieved a spacelike separation of + and −.…”

Section: Preliminariesmentioning

confidence: 99%

On the Causal and Topological Structure of the 2-Dimensional Minkowski Space

2019

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“…In the sequence of papers, [13], [15], [16], [17] and [18], the authors aim to establish a common background for the topologisation problem of a space-time. This background is the Lorentz "metric" and the structure of the lightcone, where one can define the chronological order ≪, the causal order ≺, the relation horismos → and also the chorological order <; for the last one, see in particular [17] and for a complete list of relations R depending on the lightcone see [18].…”

Section: What Is (Or Should Be) the Role Of Spacetime Topology?mentioning

confidence: 99%

Natural vs. Artificial Topologies on a Relativistic Spacetime

Papadopoulos¹

2020

Springer Optimization and Its Applications

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Consider a set M equipped with a structure * . We call a natural topology T * , on (M, * ), the topology induced by * . For example, a natural topology for a metric space (X, d) is a topology T d induced by the metric d and for a linearly ordered set (X, <) a natural topology should be the topology T < that is induced by the order <. This fundamental property, for a topology to be called "natural", has been largely ignored while studying topological properties of spacetime manifolds (M, g) where g is the Lorentz "metric", and the manifold topology T M has been used as a natural topology, ignoring the spacetime "metric" g. In this survey we review critically candidate topologies for a relativistic spacetime manifold, we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field, and give the foundations for future discussions. We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, Göbel, Hawking-King-McCarthy and others, and we examine what should be meant by the term "natural topology" for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Theorem of Gao-Wald type of "time dilation" are such examples.

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“…Each one of them is induced either from the causal or chronological orders or from (the irreflexive) horismos, with the exception of Z S which is induced by a particular spacelike non-causal order that we describe in [25]. in , respectively, at an event x.…”

Section: A Is Hausdorffmentioning

confidence: 99%

Spacetimes as Topological Spaces, and the Need to Take Methods of General Topology More Seriously

Papadopoulos

Scardigli

2019

Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

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Why is the manifold topology in a spacetime taken for granted? Why do we prefer to use Riemann open balls as basic-open sets, while there also exists a Lorentz metric? Which topology is a best candidate for a spacetime; a topology sufficient for the description of spacetime singularities or a topology which incorporates the causal structure? Or both? Is it more preferable to consider a topology with as many physical properties as possible, whose description might be complicated and counterintuitive, or a topology which can be described via a countable basis but misses some important information? These are just a few from the questions that we ask in this Chapter, which serves as a critical review of the terrain and contains a survey with remarks, corrections and open questions. AMSC: 83XX, 83F05, 85A40, 54XX keywords: Zeeman -Göbel topologies, topologising a spacetime, spacetime singularities, causal topologies, manifold topology 1 in the presence of Riemannian basic-open balls fail as well. Zeeman's main arguments against the Euclidean R 4 topology for Minkowski spacetime M (extended by Göbel for curved spacetimes) can be summarised as follows: 1. The 4-dimensional Euclidean topology is locally homogeneous, whereas M is not; every point has associated with it a light cone, separating space vectors from time vectors. 2. The group of all homeomorphisms of 4-dimensional Euclidean space is vast, and of no physical significance.Heathcote's antilogue belongs to (sic) a realist view of spacetime topology as against the instrumentalist position. A realist point of view divides the space intro structural levels, such as metric tensor field, affine connection, conformal structure, differentiable manifold and topology. Heathcote highlights that the manifold topology is present as long as the structure of manifold is present, and there are two "untenable" possibilities for a replacement of the manifold topology, in both cases by finer topologies (see [13], page 255, for more details).

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On two topologies that were suggested by Zeeman

Cited by 5 publications

References 12 publications

On the Causal and Topological Structure of the 2-Dimensional Minkowski Space

On the Causal and Topological Structure of the 2-Dimensional Minkowski Space

Natural vs. Artificial Topologies on a Relativistic Spacetime

Spacetimes as Topological Spaces, and the Need to Take Methods of General Topology More Seriously

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