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

A list of all possible causal relations in the $2$-dimensional Minkowski space $M$ is exhausted, based on the duality between timelike and spacelike in this particular case, and thirty topologies are introduced, all of them encapsulating the causal structure of $M$. Generalisations of these results are discussed, as well as their significance in a discussion on spacetime singularities.


Preliminaries.
Throughout the text, unless otherwise stated, we consider the two-dimensional Minkowski spacetime M, that is the two-dimensional real Euclidean space equipped with the characteristic quadratic form Q, where for x = (x 0 , x 1 ) ∈ M, Q(x, x) = x 2 0 − x 2 1 . We denote the light cone through an event x by C L (x), and define it to be the set C L (x) = {y : Q(y, x) = 0}. Similarly, we define the time cone as C T (x) = {y : y = x or Q(y, x) > 0} and the space cone as C S (x) = {y : y = x or Q(y, x) < 0}. We call causal cone the set C T (x) ∪ C L (x) and we observe that the event x partitions its time/light/causal cone into future and past time/light/causal cones, respectively, while it divides the space cone into − and +, respectively.
In [14] (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 −. This space-like separation is more obvious in the 2-dimensional Minkowski spacetime M. Let x ∈ M be an event. Then, we consider the future and past time-cones, C T + (x) and C T − (x), respectively, as North and South in a compass, while the space-cones C S + (x) and C S − (x), respectively, as East and West. We denote the Euclidean topology on Zeeman, in [6] (as a result of his previous work in [8]) has questioned the use of the topology E in 4-dimensional Minkowski space, as its "natural" topology, listing a number of issues, including that the Euclidean topology is locally homogeneous (while M is not) and the group of all homeomorphisms of (four dimensional) Euclidean space is of no physical significance. Zeeman proposed a topology, his "Fine" topology, under which the group of all homeomorphisms is generated by the (inhomogeneous) Lorentz group and dilatations. In addition, the light, time and space cones through a point can be deduced from this topology.
Göbel, in [7], generalised Zeeman's results for curved spacetime manifolds, and obtained that under a general relativistic frame, the Fine topology gives the significant result that a homeomorphism is an isometry. Hawking-King-McCarthy, in [12], introduced the "Path" topology, which determines the causal, differential and conformal structure of a space-time, but it was proven by Low, in [5], that the Limit Curve Theorem under the Path topology fails to hold, and so the formation of basic singularity theorems. Given that the questions that were raised by Zeeman in [6] are of a tremendous significance for problems related to the topological, geometrical and analytical structure of a spacetime, the topologisation problem for spacetimes is still open and significant.
In this article, we examine all possible (ten in number) causal relations which can appear in the 2-dimensional Minkowski spacetime and the thirty topologies which they induce.
All these topologies incorporate the causal structure of spacetime, and we believe that a generalisation to curved 4-dimensional spacetimes will equip modern problems of general relativity and cosmology with extra tools, that can be used in attempts, for example, to describe the structure of the universe in the neighbourhood of the spacetime singularities that are predicted by the singularity theorems of general relativity (ambient cosmology) or describe the description of the transition from the quantum non-local theory to a classical local theory.
2 Causal relations in the 2-Dimensional Minkowski Space.
We consider the 2-Dimensional Minkowski Spacetime M, equipped with the following relations: 1. ≪; the chronological partial order, defined as x ≪ y, if y ∈ C T + (x). We note that ≪ is irreflexive.
3. <; the chorological ("choros" is the Greek for "space", just like "chronos" is the Greek for "time") partial order, defined as x < y, if y ∈ C S + (x). We note that < is irreflexive. 4. → irr ; we define the irreflexive horismos in a similar way as we defined →, this time without permitting x to be at horismos with itself. 5. ≪ = ; we define the reflexive chronology as we defined ≪, but this time we permit x to chronologically precede itself.
6. ≺; the causal order is a reflexive partial order defined as 7. ≪ → irr ; we define the irreflexive causal order as we defined ≺, this time excluding the case that x ≺ x.
8. ≤; we define the reflexive chorology as we defined <, but this time we permit x to chorologically precede itself. 9. ≪ c ; the complement of chronological order is a reflexive partial order defined as x ≪ c y 10. < → irr ; we define the irreflexive complement of chronological order as ≪ c excluding the case that x ≪ c x. The causal automorphisms form the causality group and the acausal automorphisms form the acausality group.
The proofs of lemmas 2.1 and 2.2 can be found in [8].

Thirty Causal Topologies on the 2-dimensional Minkowski
Space.
Consider an order relation R defined on a space X. Then, consider the sets I + (x) = {y ∈ X : xRy} and I − (x) = {y ∈ X : yRx}, as well as the collections A basic-open set U in the interval topology T in (see [4]) is T The 4-dimensional Minkowski space in particular (and spacetimes in general) is not upcomplete, and a topology T in is weaker than the interval topology of [4], but for the particular case of 2-dimensional Minkowski spacetime, T in under the ten causal relations that we stated above is the actual interval topology defined in [4].
The Alexandrov topology (see [2]) is the topology which has basic open sets of the form a spacetime manifold M is strongly causal iff the Alexandrov topology is Hausdorff iff the Alexandrov topology agrees with the manifold topology.
Last, but not least, If T 1 and T 2 are two distinct topologies on a set X, then the intersection topology T int (see [10]) with respect to T 1 and T 2 , is the topology on X such that the Below, we list all possible order topologies that are generated by the ten causal relations above, either by defining the topology straight from the order (in a similar way the Alexan- . Low (see [5]) has shown that the Limit Curve Theorem fails to hold for the Path topology, and so the formation of a basic contradiction present in the proofs of all singularity theorems, fails as well (for a more extensive discussion see [11], [9] and [15]).
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. give empty set, so γ will be not a limit curve of the sequence γ n under the specified topology either 2, 3, 8, 9, 14, 15, 23, or 24 and so the Limit Curve Theorem will fail for each of these topologies. On the contrary, following the same argument, the Limit Curve Theorem will hold for each of the topologies 5, 6,11,12,17,18,20,21,26,27,29  A question that is now raised is which topology is the most appropriate one, if one can set it in this way, or the most physical one; the remark that for eight of these topologies the Limit Curve Theorem fails to hold, could bring the discussion on the need for an Ambient Cosmology to a different level. For example, the very construction of the ambient boundaryambient space model (see [16]) was an attempt to get a spacetime (the conformal infinity of an ambient space) for showing that singularities are absent and the Cosmic Cencorship becomes valid by construction. In the frame of topologies like those ones that we mentioned in this paper though, this is achieved without the need of working in extra dimensions.
Lastly, topologies 4, 10 seem to fit well in spaces consisted of girders, hypergirders and links (see [1]). Although they depend on the structure of the light cone, the question that has to be addressed is how they could be used in a description of the transition from quantum non-local theory to a classical local theory. Certainly, there is not a definite answer to this question at the present moment but we believe that methods of point-set topology will contribute significantly, as one can work using topological tools invariantly from the geometry of a spacetime.