221 research outputs found
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 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
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
-dimensional "ambient boundary") of a -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
-dimensional metric, in order to return back such a topology to its
-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 and
Let and be two given topological spaces, (respectively,
) the set of all open subsets of (respectively, ), and
the set of all continuous maps from to . We study Scott type
topologies on and we construct admissible topologies on
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
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