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.Comment: Appeared in: Mathematical Analysis, Approximation Theory and their Applications (eds T.M. Rassias and V. Gupta), Springer, New York, 2016. arXiv admin note: text overlap with arXiv:1410.595

A Few Questions on the Topologization of the Ring of Fermat Reals

This note is developing, and we will include many additions in the near future. Our purpose here is to highlight that there is plenty of space for a topological development of the Fermat Real Line.Comment: This note is developing, and we will include many additions in the near future. Our purpose here is to highlight that there is plenty of space for a topological development of the Fermat Real Lin

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.Comment: This paper has been published in the journal: Q & A in General Topology (Vol. 33, No 2, 2015.). arXiv admin note: substantial text overlap with arXiv:1302.251

On null geodesically complete spacetimes uder NEC and NGC; is the Gao-Wald "time dilation" a topological effect?

We review a theorem of Gao-Wald on a kind of a gravitational "time delay" effect in null geodesically complete spacetimes under NEC and NGC, and we observe that it is not valid anymore throughout its statement, as well as a conclusion that there is a class of cosmological models where particle horizons are absent, if one substituted the manifold topology with a finer (spacetime-) topology. Since topologies of the Zeeman-G\"obel class incorporate the causal, differential and conformal structure of a spacetime, and there are serious mathematical arguments in favour of such topologies and against the manifold topology, there is a strong evidence that "time dilation" theorems of this kind are topological in nature rather than having a particular physical meaning

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.Comment: Chapter in the Volume Topics in Mathematical Analysis and Applications, in the Optimization and Its Applications Springer Series, T. Rassias and L. Toth Eds, Springer Verlag, 2014. arXiv admin note: text overlap with arXiv:1306.214

Spacetime Singularities vs. Topologies of Zeeman-G\"obel 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-G\"obel 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 (G\"obel)? 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

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

Are four dimensions enough, a note on ambient cosmology

The group of homothetic symmetries in the conformal infinity (the $4$-dimensional "ambient boundary") of a $5$-dimensional spacetime restricts the choice of topology to a topology under which the group of homeomorphisms of a spacetime manifold is the group of homothetic transformations. Since there are such spacetime topologies in the class of Zeeman-G\"obel, under which the formation of basic contradiction present in proofs of singularity theorems is impossible, an important question is raised: why should one construct a $5$-dimensional metric, in order to return back such a topology to its $4$-dimensional conformal boundary, while such topologies, like those ones in the Zeeman-G\"obel class, are already considered as more "natural" topologies for a spacetime, rather than the artificial (according to Zeeman) manifold topology

On sliced spaces: Global Hyperbolicity revisited

We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric

Admissible topologies on C(Y,Z)C(Y,Z) and OZ(Y){\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.Comment: This paper appeared in the journal Q and A in General Topology, Volume 32, Number 1 (2014