Hausdorff separability of the boundaries for spacetimes and sequential spaces
HHausdorff separability of the boundariesfor spacetimes and sequential spaces.
J.L. Flores ∗ , J. Herrera ‡ , M. S´anchez † ∗ Departamento de ´Algebra, Geometr´ıa y Topolog´ıa,Facultad de Ciencias, Universidad de M´alaga,Campus Teatinos, 29071 M´alaga, Spain ‡ Instituto de Matem´atica e Estat´ıstica,Universidade de S˜ao Paulo,Rua do Mat˜ao, 1010, Cidade Universitaria S˜ao Paulo, Brazil † Departamento de Geometr´ıa y Topolog´ıa,Facultad de Ciencias, Universidad de Granada,Avenida Fuentenueva s/n, 18071 Granada, Spain
January 6, 2015
There are several ideal boundaries and completions in General Relativity sharingthe topological property of being sequential , i.e., determined by the convergence ofits sequences and, so, by some limit operator L . As emphasized in a classical articleby Geroch, Liang and Wald, some of them have the property, commonly regarded asa drawback, that there are points of the spacetime M non T -separated from pointsof the boundary ∂M .Here we show that this problem can be solved from a general topological viewpoint.In particular, there is a canonical minimum refinement of the topology in the comple-tion M which T -separates the spacetime M and its boundary ∂M —no matter thetype of completion one chooses. Moreover, we analyze the case of sequential spacesand show how the refined T -separating topology can be constructed from a modifi-cation L ∗ of the original limit operator L . Finally, we particularize this procedure tothe case of the causal boundary and show how the separability of M and ∂M can beintroduced as an abstract axiom in its definition. Keywords: boundaries of spacetimes, causal, conformal, geodesic and bundle boundary,Hausdorff separability, sequential space, limit operator.
MSC:
Primary 53C50, 83C75, Secondary: 54D55, 54A20. a r X i v : . [ m a t h - ph ] J a n ontents -separating boundaries in the general and sequential cases 8 T -separability and c-boundary 14 D -separability as an admissibility condition . . . . . . . . . . . . . . . 21 A recurrent topic in the mathematical study of relativistic spacetimes, is the definition of sometypes of ideal boundaries which encode relevant information about them. Being this a naturalpractice from a mathematical viewpoint, there are further physical motivations coming from theholographic principle and the anti-de Sitter/conformal field theory correspondence. There havebeen quite a few of proposals of boundaries (including geodesic, bundle, abstract, conformaland causal ones, see Section 4), each one with its own limitations. One of the most commonproblems appears from the topological viewpoint, as typically one may find that some point ofthe spacetime M is not topologically well separated (i.e., non-Hausdorff separated or even non-T -separated) from some point of the boundary ∂M . Such a property seems so undesirable, thathas been considered as a reason to reject a priori many possible boundaries [15]. The purposeof this paper is to reconsider this question from a broad mathematical perspective.In a naive approach to the problem, one would like that a boundary ∂M satisfied: (i) itinherits the good topological properties of the topology of the spacetime M , and (ii) it recordsinformation of missing points, that is, it would satisfy, for example: if a point of the spacetimeis removed, the boundary will identify this unequivocally and will allow to restore the missingpoint. However, a closer look shows that these requirements would be too restrictive. About therequisite (i), notice first that one can find an open subset D of Euclidean space R n with, say,a fractal topological boundary ∂D ⊂ R n . A worse contradiction with (i) appears when some ofthe commented relativistic boundaries yield naturally a couple of non T -separated boundarypoints: this becomes a natural consequence of the fact that there is no any distance associated toa spacetime . About (ii), assume that one is considering a boundary invariant by homotheties. When a distance exists on M (as happens, for example, when M is endowed with a -positive definite- Riemannianmetric) one may expect that it will be extensible to the boundary ∂M and, thus, ∂M would be Hausdorff. R n \{ } and R n \ ¯ B (0 , p ∈ M and some Q ∈ ∂M , appears naturally when the requirements (i) and (ii) fail: by the failure of the former,one can admit the bad topological separation of two boundary points P, Q ∈ ∂M but, because ofthe failure of the latter, one might discover that P admits an interpretation as a missing point p for some new spacetime. Roughly, our answer to this problem will be the following: No matter how the completion M = M ∪ ∂M is constructed, one can define a uniqueminimal refinement of the topology (i.e. a finer topology) in M which will T -separatethe points of the manifold M and those of the boundary ∂M . So (whenever thespacetime M is clearly identified in the completion M ), this refined completion wouldnot be rejected a priori by claiming pathological separability properties. Moreover, theminimal character of the refinement will preserve most of the desirable properties ofthe original topology. Once this aim is carried out, our next aim is motivated by the following observation. Someof the previous topologies are defined in terms of a limit operator L which characterizes theconvergence of the sequences. Thus, it is interesting to study especially this case and, concretely,how to construct a Hausdorff separating topology by modifying directly L . Mathematically, thespaces whose topology can be characterized by such an operator are the sequential spaces . Thisis a vast class of spaces (it includes all the first countable ones, in particular the metric spaces,see Section 3). So, we will also carry out a study of independent mathematical interest aboutthe refinement process for sequential spaces endowed with a limit operator.As a last aim, this study will be applied specifically to the causal boundary (c-boundary forshort in the remainder) of spacetimes. The reason is twofold. On the one hand, this boundaryhas been widely studied recently, because it seems the unique intrinsic and general alternative tothe Penrose conformal boundary (the latter is commonly used in Mathematical Relativity, butits existence is ensured only in quite particular cases). On the other hand, the redefinition ofthe c-boundary in [9] left open the possibility of new refinements of such a notion, by imposingadditional properties to the boundary which might be judged as desirable a priori. The propertyof separation studied here is a good example of this possibility.This article is organized as follows. In Section 2 we recall some general features of sequentialspaces, and prove some first properties about limit operators (Subsection 2.1, specially Propo-sitions 2.5 and 2.7). The following subtlety is emphasized. A limit operator L for a topology τ is full or of first order when L provides directly all the limits of all the convergent sequences for τ ; in this case, it is univocally determined and can be denoted by L τ . Otherwise, L determinesonly some limits of convergent sequences, which are enough to determine the topology. Evenmore, one can wonder if L is of second order (roughly, the iteration of L twice is enough to deter-mine L τ ) or of other higher orders; this is briefly explained in Subsection 2.2. Even though anytopology (sequential or not) admits a unique first order L τ , it is interesting to consider also thecase of non-first order limit operators because: (a) the topology of some relativistic boundaries isdefined by imposing that naturally, some sequences must converge ; however, such a convergence But recall that even a non-symmetric distance, as the one associated to a non-reversible Finsler metric, maylose properties when extended to the boundary. In fact, the Cauchy completion of a Finsler manifold may notbe T , see [11, Section 3]. This is relevant for the case of spacetimes too, as Finsler metrics appear naturallywhen the conformal class of simple classes of spacetimes are considered [6, 11]. L (as, eventually, those required for our refinement of the topologies) might give limit operatorsthat are not of first order, even if L was.Section 3. In Subsection 3.1, the problem of T -separating a domain D of its complement X \ D , in an arbitrary topological space ( X, τ ) (see Defn. 3.1), is considered. A (minimum)unique explicit topology τ ∗ is shown to solve the problem (Theorem 3.3). In Subsection 3.2,we consider the problem for the special case of sequential spaces. Recall that for these spaces,it is natural to wonder about a separating topology which is minimum among the sequentialones . The existence of a unique sequential topology τ ∗ Seq solving this problem of T -separationis also obtained (Theorem 3.5). Finally, in Subsection 3.3 we analyze the role of the chosenlimit operator L for the case of sequential topologies. Concretely, the domain D allows to definenaturally a modification L ∗ of the original operator L (see formula (8)). When L is of firstorder, the topology generated by L ∗ can be identified with the topology previously obtained (seeTheorem 3.9 and Figure 1; notice that L ∗ is then necessarily of first order too); otherwise, someother properties still hold (Proposition 3.7).Section 4. In Subsection 4.1 we review briefly the most common boundaries for spacetimes,and explain how previous results can be applied to ensure the Hausdorff separability betweenpoints of M and ∂M . Special care is put in the discussion of a general example by Geroch, Liangand Wald [15] which shows that, under general hypotheses, the g-boundary and other boundariesintroduced in General Relativity will have a pair of non- T related points, one of them in thespacetime and the other in the boundary. Our previous results allow to circumvent this problemand suggest that the g-boundary should be redefined. In the remainder we focus in the case ofthe c-boundary, whose topology (the chronological topology τ chr ) is defined as the one derivedfrom some limit operator L chr . In Subsection 4.2, a short review of the causal completion isprovided. In Subsection 4.3 we check that the good properties of this completion (previouslysummarized in Theorem 4.3) are maintained when one uses both, the corresponding separatingtopology τ ∗ chr and the sequentially separating one ( τ ∗ chr ) Seq (Theorem 4.4).Section 5. Here, we analyze the process to define the c-boundary from very general admissibilityproperties, according to [9]. In Subsection 5.1, the admissibility conditions in [9] are revisited,obtaining an extension of the results in that reference which include non-necessarily sequentialtopologies (Theorem 5.4). In Subsection 5.2, the following aim is achieved. Assume that theHausdorff separability between points of M and ∂M is incorporated as one of these admissibilityproperties that characterize the c-boundary. Then the corresponding topology becomes equalto the chronological topology refined in the general D -separating way explained in Section 3,under the hypothesis that L chr is of first order and even under less restrictive hypothesis whichexhibit the accuracy of the approach (see Theorem 5.8, the discussion in Remark 5.9 and thesummarizing Figure 3).The article finishes with an Appendix containing examples which show that most of the sub-tleties suggested by our procedures can occur effectively. In this section we develop some properties about limit operators and sequential spaces. There is awell established literature on the later, see [12, 16] and references therein for general background.
Let X be an arbitrary set, let S ( X ) the set of all sequences in X and P ( X ) the set of parts of X . Following [9, Sect. 3.6], [11, Sect.5.2.2] we consider:4 efinition 2.1 A map L : S ( X ) → P ( X ) is a limit operator if it satisfies the following com-patibility condition for subsequences : L ( σ ) ⊂ L ( κ ) , for any σ, κ ∈ S ( X ) with κ ⊂ σ . (1) In this case, the topology derived from L is the topology τ L whose closed sets are those subsets C of X such that L ( σ ) ⊂ C for any sequence σ ⊂ C . In fact, the compatibility condition for subsequences allows to prove easily that the so-definedclosed subsets satisfy the axioms for a topology.
Remark 2.2 (1) We can assume with no loss of generality that any limit operator is coherent in the sense that, for any x ∈ X one has x ∈ L ( { x n = x } ). This will be assumed explicitly inthe case of limit operators of k -th order below.(2) If a topology τ on X has been prescribed, one can define a unique associated limit operator L τ which gives directly the convergence of sequences in ( X, τ ) (i.e., such that the converse offormula (3) also holds), namely: L τ : S ( X ) → P ( X ) , L τ ( σ ) := { p ∈ X : σ → p with τ } . (2)(3) For an arbitrary limit operator L , the derived topology satisfies clearly the following im-plication: p ∈ L ( σ ) = ⇒ σ → p with the topology τ L . (3)However, the converse does not hold in general and we will be also interested in this possibility.A trivial (non-coherent) example is just the limit operator L ( σ ) = ∅ for all σ ∈ S ( X ), whosederived topology is the discrete one. A coherent example can be constructed as follows. Let( X, τ ) be any sequential topological space with a convergent sequence σ = { x n } → x ∞ such thatno subsequence of σ is equal to σ , except σ itself (for example, the elements of σ could be alldistinct as in the case ( X, τ ) = R , σ = { /n } , x ∞ = 0). Let L τ be the associated limit operatorand define a new limit operator L such that L (˜ σ ) = ∅ , where ˜ σ is any sequence that contains σ as a subsequence, and L ( µ ) = L τ ( µ ) for any other sequence µ . As µ can be chosen equal to anysubsequence of σ (except σ itself), this implies that, in any case, σ will converge to x ∞ with τ L (cid:48) and, then, τ L (cid:48) = τ .Obviously, these examples correspond to limit operators where L is regarded as empty on asequence in a highly arbitrary way. Nevertheless, one may find naturally the following situationin the case of non-Hausdorff topologies: the limit operator yields a infinite set of limits, andthese limits will yield naturally more limits. This will lead to the notion of k -th limit operator,to be studied below.The considerations above motivate the following definition. Definition 2.3
Let D be a subset of X . We will say that a limit operator L on X is of firstorder on D if, for any sequence σ ⊂ D and any p ∈ D : p ∈ L ( σ ) ⇐⇒ σ → p with τ L . When L is of first order on D = X (i.e. L = L τ L ) then we say simply that L is of first order . As a first simple property to be used in the remainder, we have the following.
Lemma 2.4
Let
L, L (cid:48) be two limit operators on X with derived topologies τ L and τ L (cid:48) resp. If L (cid:48) ( σ ) ⊂ L ( σ ) for any sequence σ ⊂ X , (4) then τ L ⊂ τ L (cid:48) . Moreover, when L is of first order, the converse also holds. roof. Let C be a closed set for τ L , that is, L ( σ ) ⊂ C for any sequence σ ⊂ C . As L (cid:48) ( σ ) ⊂ L ( σ ) ⊂ C , we deduce that C is also closed for τ L (cid:48) , and so, τ L ⊂ τ L (cid:48) .For the converse, assume that L is of first order and q ∈ L (cid:48) ( σ ) for some sequence σ ⊂ X .Then, σ → q for τ L (cid:48) , and, by the inclusion τ L ⊂ τ L (cid:48) , this limit also holds for τ L . As L is of firstorder, necessarily q ∈ L ( σ ), as required.Recall that a topological space ( X, τ ) is called sequential if any sequentially closed set (i.e. aset which contains all the limits of any sequence contained in it) is also a closed set. Note thatthe converse is true in any topological space.
Proposition 2.5
Let X be a set:(a) For any limit operator L on X , its derived topology τ L is sequential.(b) For any topology τ on X , its associated limit operator L τ (see formula (2)) has a derivedtopology τ Seq := τ L τ , which is the coarsest one among the sequential topologies containing τ .Moreover, L τ is a limit operator of first order for τ Seq .(c) A topology τ on X is sequential if and only if τ = τ Seq .Proof. (a) is straightforward from the definitions.For (b), the inclusion τ ⊂ τ Seq follows from the fact that closed subsets are sequentially closedfor any topology. Let τ (cid:48) be another sequential topology with τ ⊂ τ (cid:48) . If σ converges to p with τ (cid:48) , then σ → p with τ , which implies p ∈ L τ ( σ ). Hence, sequentially closed sets for τ Seq are alsosequentially closed for τ (cid:48) , and so, τ Seq ⊂ τ (cid:48) . For the first order character of L τ , assume that σ converges to p with τ Seq . Since τ ⊂ τ Seq , the sequence σ must converge to p with τ , and thus, p ∈ L τ ( σ ).Finally, the right implication of (c) follows from the assumption that sequentially closed for τ implies closed (to the left, use (a)). Remark 2.6
Summing up, a topological space (
X, τ ) is sequential if and only if τ = τ L for somelimit operator L on X , which can be chosen of first order (and, then, equal to L τ ). From theprevious properties and the known ones about sequential spaces (see [16], especially its figures1.1 and 1.3) the following observations are in order:(1) The map τ (cid:55)→ L τ from the set T ( X ) of all the topologies on X to the set of all the firstorder limit operators on X , is an onto map (as L = L τ L ). Nevertheless, it is not one to one, asany non-sequential topology τ has associated the limit operator L τ equal to the limit operatorfor the sequential topology τ Seq . In fact, one has then an onto map T ( X ) → T Seq ( X ) , τ (cid:55)→ τ Seq where T Seq ( X ) is the set of all the sequential topologies on X .(2) Sequential spaces generalize Fr´echet-Uryshon spaces (namely, the closure of a set consistsof the limits of all the sequences in that set) and, then, first countable spaces, which includeall the metrizable ones; however, sequential spaces can be regarded as quotients of metrizablespaces.If X is a sequential space, the continuity of a map f : X → Y into a topological space Y canbe characterized by the preservation of the limits of sequences. Nevertheless, the closure of a set A ⊂ X may contain strictly the set of all the limits of sequences in A (in contrast with Fr´echet-Uryshon), the uniqueness of the limits of sequences does not imply Hausdorffness (in contrastwith first countability), and sequential compactness (i.e., the property that any sequence admitsa convergent subsequence) plus Hausdorffness and first countability, do not imply compactness(in contrast with metrizability). Nevertheless, for sequential spaces, sequential compactnessbecomes equivalent to countable compactness (see for example [16, Proposition 3.2]) and, as thelatter is weaker than compactness, compact sequential spaces are sequentially compact .6he property of being sequential is not inherited by arbitrary subspaces, but it is inherited byopen (or closed) subspaces [12, Prop. 1.9]. More sharply, the limit operators can be inherited byopen subspaces, in the following sense. Proposition 2.7
Let D be an open subset of a sequential space ( X, τ L ) . The topology τ L | D induced on D by τ L coincides with the topology τ L | D derived from the restricted limit operator L | D given by L | D ( σ ) := L ( σ ) ∩ D, for any sequence σ ⊂ D .Proof. It is straightforward to check that L | D is a limit operator if so is L . So, we focus on theidentification between τ L | D and τ L | D .In order to prove that τ L | D ⊂ τ L | D , let C be a closed set of τ L | D . Then, C = C (cid:48) ∩ D with C (cid:48) aclosed set of τ L . Let σ be any sequence in C . As σ ⊂ C ⊂ C (cid:48) and C (cid:48) is closed for τ L , necessarily L ( σ ) ⊂ C (cid:48) . Therefore, L | D ( σ ) = L ( σ ) ∩ D ⊂ C (cid:48) ∩ D = C, and thus, C is closed for τ L | D .For the inclusion τ L | D ⊂ τ L | D , let B ⊂ D an open set of τ L | D . It is enough to prove that B isalso open for τ L . So, assume by contradiction that X \ B is not closed for τ L , and thus, thereexist a sequence σ ⊂ X \ B and a point p ∈ B ⊂ D such that p ∈ L ( σ ). As D is open for τ L , σ ⊂ D eventually. In particular, p ∈ L ( σ ) ∩ D = L | D ( σ ), and from (3), σ converges to p with τ L | D . But B is open for τ L | D , hence σ eventually intersects B , which is a contradiction. When a limit operator L is not of first order, L ( σ ) does not provide all the topological limits ofsome sequence σ ⊂ X , that is, L ( σ ) (cid:32) L τ L ( σ ). Nevertheless, in view of the discussion in Remark2.2 (as well as Example 6.2 below), it is natural to wonder when any point in L τ L ( σ ) \ L ( σ )( (cid:54) = ∅ )can be obtained by iterating successively the limit operator L . In order to formalize this idea,we will assume that L is a coherent limit operator (Remark 2.2 (1)) and consider the followingtransfinite definition: L ( σ ) := L ( σ ) , L i ( σ ) := { p ∈ X : p ∈ L ( { p n } ) , { p n } ⊂ ∪ j
The inclusion L i ( σ ) ⊂ L τ L ( σ ) holds for any σ ⊂ X and any ordinal i .Proof. One needs to show that if p ∈ L i ( σ ) then σ converges to p with the topology τ L , i.e., if σ = { p n } and p ∈ U for some open set U of τ L , there exists n such that p n ∈ U for all n ≥ n .The proof will follow by transfinite induction. For i = 1, the result is known from (2). So,assume that it is true for all j < i , and let us prove it for i . From the definition of L i ( σ ),there exists a sequence { q n } ⊂ ∪ j
If a coherent limit operator L is of k -th order then L k ( σ ) = L i ( σ ) for any σ ⊂ X and any ordinal i ≥ k . -separating boundaries in the general and sequential cases In this section we consider the following problem. Let (
X, τ ) be a topological space, and let D ⊂ X be an open subset with some good topological properties, namely, Hausdorff and locallycompact. Typically, D will be a Lorentzian manifold and X some topological completion, so that D is dense in X . However, we do not impose a priori the density, allowing X \ D to be a bigger“crust”. Now, assume that there is some pair of points p ∈ D , q ∈ X \ D that are not Hausdorffseparated, and we look for a “minimal refinement of the topology around ∂D ” in order to ensurethat such pairs are T -separated. Formally, we are looking for a new topology τ ∗ which satisfiesthe following requirements. Definition 3.1
Let ( X, τ ) be a topological space, and let D ⊂ X be an open subset which isHausdorff and locally compact. A topology τ ∗ is minimally D -separating if it satisfies:(A Fin ) Refinement of τ : the topology τ ∗ is finer than τ , i.e. τ ⊂ τ ∗ . (A Sep ) T -separability of points of D and X \ D : for any p ∈ D , q ∈ X \ D , there exist U ∈ τ and V ∈ τ ∗ such that p ∈ U, q ∈ V and U ∩ V = ∅ .(A Min ) Minimality : τ ∗ is a minimal topology among those satisfying (A Fin ) and (A
Sep ), i.e. noother topology satisfying conditions (A
Fin ) and (A
Sep ) is strictly coarser than τ ∗ . Remark 3.2 (1) From property (A
Fin ), the subset D will be also open for τ ∗ .(2) The condition of minimality (A Min ) is essential in order to avoid trivial topologies. Forexample, if we considered as τ ∗ the one generated by τ and the crust X \ D as a subbasis, then τ ∗ would satisfy (A Fin ) and (A
Sep ), but at the cost of a separability which would forbid any pointin the τ -boundary ∂D to be τ ∗ -continuously reachable from D (i.e., the τ ∗ -boundary of D wouldbe the empty set); a similar drawback would happen if one considered the topology generatedby τ and the sets { q } for every point q ∈ X \ D which is not T -separated from some point in D .(3) There exists a small asymmetry in the condition of T -separability (A Sep ), as the separatingneighborhood U is not only required to belong to τ ∗ but also to τ . Notice that only D is assumedto be a topological subspace with additional nice properties (Hausdorfness, local compactness).This asymmetry in (A Sep ), plus the minimality in (A
Min ), will ensure that the topology τ ∗ preserves most of the original properties of ( X, τ ). Among them, τ and τ ∗ will induce the sametopology on D (see Theorem 3.3 below). If the asymmetry were not imposed, τ | D (cid:54) = τ ∗ | D mayoccur (Example 6.3).Now, we can state the following general result for topological spaces.8 heorem 3.3 Let ( X, τ ) be a topological space and D ⊂ X be a Hausdorff locally compact opensubset of X . The topology τ ∗ generated by the subbasis S = τ ∪ { X \ K : K is a compact subset of D } (7) is the unique topology which satisfies the properties (A Fin ), (A
Sep ) and (A
Min ) above, i.e., theunique minimally D -separating topology. Moreover, the restrictions of τ ∗ and τ on D coincide.Proof. Clearly, τ ∗ satisfies (A Fin ). In order to prove (A
Sep ), consider p ∈ D and q ∈ X \ D . Let K be a compact neighborhood of p in D and U its interior. From the definition of the subbasis S in (7), V := X \ K is open for τ ∗ and, so, U and V are the required open sets.In order to prove (A Min ) and the uniqueness, we will show that τ ∗ ⊂ τ (cid:48) for any other topology τ (cid:48) satisfying (A Fin ) and (A
Sep ). Since τ ⊂ τ (cid:48) , it suffices to show that X \ K ∈ τ (cid:48) for any compactsubset K of D . So, let us prove that X \ K is a neighborhood of any q ∈ X \ K . From theproperty (A Sep ) and the Hausdorffness of D , for every p ∈ K there exists U p ∈ τ ⊂ τ (cid:48) and V p ∈ τ (cid:48) such that p ∈ U p , q ∈ V p and U p ∩ V p = ∅ . Since K is compact, there exists { U p i } ni =1 ⊂ { U p } p ∈ K such that K ⊂ ∪ ni =1 U p i . Then, V := ∩ ni =1 V p i ∈ τ (cid:48) satisfies q ∈ V and V ∩ K ⊂ V ∩ ( ∪ ni =1 U p i ) = ∅ .Therefore, X \ K is a neighborhood of q , as required.For the last assertion, it suffices to show that A ∩ D ∈ τ for any A ∈ S (recall that S issubbasis of τ ∗ ). So, assume that A = X \ K , where K is a compact subset of D (if A ∈ τ ,the conclusion follows trivially from D ∈ τ ). Since D is Hausdorff, K is closed in D , and thus, D \ K (= A ∩ D ) ∈ τ , as required. Next, we consider the same problem as before but restricting our attention to sequential spaces.To do that, first we have to introduce the following adapted version of Definition 3.1:
Definition 3.4
Let ( X, τ (= τ L )) be a sequential topological space, and let D ⊂ X be an opensubset which is Hausdorff and locally compact. A topology τ ∗ Seq is sequentially minimally D -separating if it satisfies (A Fin ) and (A
Sep ) from Definition 3.1, and the following minimalitycondition:(A
SeqMin ) Sequential minimality: τ ∗ Seq is a minimal topology among the sequential ones satisfying(A
Fin ) and (A
Sep ). Now, we can state the following result for sequential topological spaces:
Theorem 3.5
Let ( X, τ (= τ L )) be a sequential topological space, and let D ⊂ X be a Hausdorfflocally compact open subset of X . The topology ( τ ∗ ) Seq (where τ ∗ is the unique minimally D -separating topology according to Theorem 3.3 and the notation introduced in Proposition 2.5(b) isused), is the unique sequentially minimally D -separating topology, that is. τ ∗ Seq exists, it satisfies(A
SeqMin ) as an unique minimum and ( τ ∗ ) Seq = τ ∗ Seq . Moreover, the restrictions of τ ∗ Seq and τ coincide on D .Proof. From Proposition 2.5 (b), the topology ( τ ∗ ) Seq is the coarsest topology among thesequential ones containing τ ∗ . So, ( τ ∗ ) Seq clearly satisfies conditions (A
Fin ) and (A
Sep ), as theseconditions are already satisfied by τ ∗ . Moreover, any other sequential topology τ (cid:48) satisfying(A Fin ) and (A
Sep ), must contain τ ∗ . So, from the coarsest character of ( τ ∗ ) Seq , the topology τ (cid:48) must contain ( τ ∗ ) Seq . Therefore, we deduce both, ( τ ∗ ) Seq satisfies condition (A
SeqMin ) and it isunique. 9he last assertion follows from Proposition 2.7 and the fact that, for all p ∈ D and all sequence σ ⊂ D , the following chain of equivalences holds p ∈ L τ ( σ ) ⇐⇒ σ converges to p with τ ⇐⇒ σ converges to p with τ ∗ ⇐⇒ p ∈ L τ ∗ ( σ ) , where the first equivalence follows as L τ is the (first order) limit operator associated to τ , thesecond one from the last assertion of Theorem 3.3 and the third one from the definition of L τ ∗ .This result provides an elegant solution to our problem. However, one can wonder for a betterunderstanding of τ ∗ Seq . Notice that the topology τ ∗ Seq is defined in terms of the limit operator L τ ∗ , which is not directly based on some limit operator L of τ , but on the topology τ ∗ . This isa difficulty in order to manage such a topology for practical purposes. This subsection is devoted to find an alternative limit operator L ∗ , directly constructed from L , whose derived topology τ L ∗ coincides with the sequentially minimally D -separating one τ ∗ Seq .The natural candidate is: L ∗ ( σ ) := (cid:26) L ( σ ) ∩ D if ∃ κ ⊂ σ and p ∈ D such that p ∈ L ( κ ) L ( σ ) otherwise (8)(so, L ∗ ( σ ) = L ( σ ) ∩ D not only when L ( σ ) ∩ D (cid:54) = ∅ but also when L ( κ ) ∩ D (cid:54) = ∅ for somesubsequence κ ; this is necessary in order to ensure that L ∗ is a limit operator, see Proposition3.7). Indeed, the definition of L ∗ suggests that it is the smallest modification of L such thatno sequence σ ⊂ X will converge to both, p ∈ D and q ∈ X \ D . However, a caution must betaken into account: this property will be ensured only if L ∗ is of first order —otherwise, σ might τ L ∗ -converge to both p and q , even if q (cid:54)∈ L ( σ ). In order to ensure that L ∗ is of first order,a natural requirement will be to assume that so is L . Of course, this is not restrictive froma fundamental viewpoint, as one can always replace it by the associated limit operator L τ in(2), which is of first order. However, this caution must be taken into account from a practicalviewpoint, when L ∗ is being computed from a sequential topology τ constructed by means ofsome prescribed limit operator L (as in the case of some commented relativistic boundaries).With this aim for L ∗ , notice first that formula (4) in Lemma 2.4 can be regarded as a charac-terization of the property (A Fin ) of Defn. 3.1 in terms of the limit operator L . It is convenientto rewrite also the condition (A Sep ) in such terms as follows.
Lemma 3.6
Let
L, L (cid:48) be two limit operators on X with derived topologies τ L and τ L (cid:48) resp., andlet D be open, locally compact and Hausdorff for τ L . Assume also that τ L ⊂ τ L (cid:48) .If the topology τ L (cid:48) satisfies the condition (A Sep ) (with τ ∗ ≡ τ L (cid:48) , τ ≡ τ L ), then one has:when L (cid:48) ( σ ) ∩ D (cid:54) = ∅ for some σ ⊂ X , then L (cid:48) ( σ ) ⊂ D . (9) The converse is true if we assume that L is of first order on D and, additionally: L | D ( σ ) = L (cid:48) | D ( σ ) for any sequence σ ⊂ D (10) (so that τ L | D = τ L (cid:48) | D by Proposition 2.7).Proof. To the right, observe that if (9) does not hold, then there exist p ∈ D, q ∈ X \ D and asequence σ ⊂ X such that p, q ∈ L (cid:48) ( σ ). Then, from (3), σ converges with τ L (cid:48) to both p, q , whichcontradicts that τ L (cid:48) satisfies (A Sep ). 10or the converse, for each p ∈ D , take some compact (for τ L of, equally by (10), for τ L (cid:48) )neighborhood K ⊂ D of p , and let U ∈ τ L be its τ L -interior. The property (A Sep ) will followtrivially if we prove that X \ K is open for τ L (cid:48) , that is, if L (cid:48) ( σ ) ⊂ K for any sequence σ ⊂ K . So,assume by contradiction that q ∈ L (cid:48) ( σ ) \ K for some sequence σ ⊂ K . Then σ → q with τ L (cid:48) and,therefore, with τ L . Note that K is a compact subset of D which contains σ , and both, D and K are sequential spaces with the restriction of τ L (use either [12, Prop. 1.9] or Proposition 2.7for the sequentiality of D and, then, [16, Lemma 3.7] for K ). So, the compactness of K impliesits sequential compactness (see the end of Remark 2.6) and there exists a subsequence κ ⊂ σ and some p (cid:48) ∈ K ⊂ D such that κ → p (cid:48) with τ L . Since L is of first order on D , necessarily p (cid:48) ∈ L ( κ ) which, combined with (10), implies p (cid:48) ∈ L (cid:48) ( κ ). On the other hand, by hypothesis, q ∈ L (cid:48) ( σ ) \ K . So, there are two possibilities: either q ∈ D , which contradicts the Hausdorffnessof τ L (cid:48) | D = τ L | D , or q ∈ X \ D , which contradicts property (9) applied to κ .The hypotheses under which the condition (9) characterizes (A Sep ) are technical, but they willbecome natural in the applications of Lemma 3.6. Next, let us characterize L ∗ for any limitoperator L . Proposition 3.7
Let ( X, τ L ) be a sequential topological space and D ⊂ X a Hausdorff locallycompact open subset of X . Then, L ∗ is a limit operator which satisfies the properties (4), (9)and (10), and it is the maximum operator satisfying them, that is: if L (cid:48) is another limit operatorsatisfying (4), (9) and (10), then L (cid:48) ⊂ L ∗ (i.e., L (cid:48) ( σ ) ⊂ L ∗ ( σ ) for any sequence σ ⊂ X ).Proof. First, let us prove that L ∗ is a limit operator, i.e. it satisfies (1). Assume by contradictionthat there exist a sequence σ and a subsequence κ such that L ∗ ( σ ) (cid:54)⊂ L ∗ ( κ ). Recall that L ∗ ( σ ) ⊂ L ( σ ) ⊂ L ( κ ). Hence, necessarily L ∗ ( κ ) (cid:54) = L ( κ ). In particular, L ( κ ) ∩ D (cid:54) = ∅ , and thus, L ∗ ( σ ) = L ( σ ) ∩ D ⊂ L ( κ ) ∩ D = L ∗ ( κ ), in contradiction with the initial hypothesis.From the definition of L ∗ , it satisfies the properties (4), (9) and (10). So, it remains to provethe maximal character of L ∗ . Let L (cid:48) be another limit operator satisfying such properties. Inorder to prove that L (cid:48) ( σ ) ⊂ L ∗ ( σ ) for any sequence σ ⊂ D , consider the cases: • There exists κ ⊂ σ and p ∈ D such that p ∈ L ( κ ). Observe that, as D is open, we canassume κ ⊂ D . Then, L ∗ ( σ ) = L ( σ ) ∩ D from the definition (8) and p ∈ L (cid:48) ( κ ) as L (cid:48) satisfies (10). Assume by contradiction that L (cid:48) ( σ ) (cid:54)⊂ L ∗ ( σ ). There exists q ∈ L (cid:48) ( σ ) suchthat q (cid:54)∈ L ∗ ( σ ) and, necessarily q (cid:54)∈ D by (10). So, q belongs to both L (cid:48) ( κ ) and X \ D ,which absurd as p ∈ L (cid:48) ( κ ) ∩ D and L (cid:48) satisfies (9). • There are no κ ⊂ σ under previous conditions. Then, L ∗ ( σ ) = L ( σ ), and, from (4), onehas L (cid:48) ( σ ) ⊂ L ( σ ) = L ∗ ( σ ).The following technical assertion will be useful; notice that Lemma 3.6 gives conditions for itsapplicability. Lemma 3.8
Assume that τ L ∗ satisfies (A Sep ). If σ → q with τ L ∗ and q ∈ L ( σ ) , then q ∈ L ∗ ( σ ) .Proof. If q ∈ D then q ∈ L ( σ ) ∩ D = L ∗ ( σ ), as required. So, assume that q ∈ X \ D . It sufficesto prove that L ( κ ) ∩ D = ∅ for any subsequence κ ⊂ σ (since then q ∈ L ( σ ) = L ∗ ( σ )). Assumingby contradiction that there exists p ∈ L ( κ ) ∩ D (cid:54) = ∅ , the sequence κ converges to both, p and q ,with τ L ∗ , in contradiction with property (A Sep ).Now, we can establish the following result that clarifies the role of limit operators:
Theorem 3.9
Let ( X, τ ≡ τ L ) be a sequential topological space and let D ⊂ X be a Hausdorfflocally compact open subset of X . Assume that the limit operator L is of first order on D . Then,the following statements hold: i) The topology derived by the limit operator L ∗ defined on (8) is finer than the unique se-quentially minimally D-separating topology. So, we have the following chain of topologies: τ ⊂ τ ∗ ⊂ τ ∗ Seq ⊂ τ L ∗ . (11) (ii) If L is of first order (on all X ), then L ∗ is also of first order and both topologies τ ∗ Seq and τ L ∗ coincide.Proof. For (i), let us consider a set C which is closed set for the topology τ ∗ Seq and prove thatsuch a set is also closed for τ L ∗ . Our aim is to show that L ∗ ( σ ) ⊂ C for all sequence σ ⊂ C .Consider a point p ∈ L ∗ ( σ ) and recall that, from the definition of L ∗ , we may consider twocases. For the first one, assume that L ∗ ( κ ) ∩ D (cid:54) = ∅ for some subsequence κ ⊂ σ , and so, that p ∈ L ∗ ( σ ) = L ( σ ) ∩ D . Then, σ converges to p with τ and σ will abandon any compact set of D not containing the point p (recall that D is Hausdorff). Therefore, σ converges to p with τ ∗ and, from the definition of its limit operator, p ∈ L τ ∗ ( σ ). But τ ∗ Seq = ( τ ∗ ) Seq (Theorem 3.5) and C is closed for τ ∗ Seq , so p ∈ C .For the second case, assume that L ( κ ) ∩ D = ∅ for all subsequence κ ⊂ σ , and so, p ∈ X \ D .As p ∈ L ∗ ( σ ), we also have that p ∈ L ( σ ), and so, σ converges to p with τ . Moreover, sucha sequence will converge to p also with τ ∗ : otherwise, there would be a compact K and asubsequence κ ⊂ K of σ converging to a point q ∈ D with the topology τ . However, as L is offirst order on D , this would imply that q ∈ L ( κ ) ∩ D (cid:54) = ∅ , a contradiction. So, σ converges to p with τ ∗ and, thus, p ∈ L τ ∗ ( σ ). In conclusion, as C is closed for τ ∗ Seq , again p ∈ C , as desired.To finish (i), recall that the two first inclusions in (11) are obvious from (A Fin ) in Defn. 3.1 andTheorem 3.5 (notice also Prop. 2.5 (b)), respectively.Now, let us prove (ii). For the first order character of L ∗ , observe first that since L ∗ satisfiesproperties (4), (9) and (10) (recall Proposition 3.7), and L is of first order, Lemmas 2.4, 3.6,ensure that τ L ∗ satisfies (A Fin ) and (A
Sep ). Now assume that σ → q with τ L ∗ for some sequence σ ⊂ X . From (A Fin ), σ → q with τ L and, as L is of first order, q ∈ L ( σ ). Then, Lemma 3.8applies, and q ∈ L ∗ ( σ ) follows, as desired.For the second assertion of (ii), recall that from (i), we have the inclusion τ ∗ Seq ⊂ τ L ∗ . As L τ ∗ is a first order limit operator, Lemma 2.4 ensures that L ∗ ( σ ) ⊂ L τ ∗ ( σ ) for all sequence σ ⊂ X .So, taking into account Proposition 3.7, it suffices to prove that L τ ∗ satisfies properties (4), (9)and (10). Properties (4) and (9) follow from properties (A Fin ) and (A
Sep ) (recall Lemmas 2.4,3.6) while (10) is a consequence of Theorem 3.5.
Remark 3.10
As announced at the beginning of the present subsection, the hypotheses of being L of first order is necessary in Theorem 3.9 (even though it is not in Proposition 3.7), but this isnot a fundamental restriction because, given a sequential topology τ , one can consider its (firstorder) limit operator L τ given in (2).Nevertheless, Theorem 3.9 allows to understand better the role of L and L ∗ in order to obtainthe separating topologies. Summing up, associated to a sequential topology τ writtten as τ = τ L we have defined three different topologies: τ ∗ , τ ∗ Seq (= ( τ ∗ ) Seq ) and τ L ∗ . They always satisfy τ ⊂ τ ∗ ⊂ τ ∗ Seq and, whenever τ L ∗ satisfies (A Sep ) (in particular, when L is of first order on D ), τ ∗ Seq ⊂ τ L ∗ . Thefirst constructed topology τ ∗ satisfies the required separating properties, but it is not necessarilysequential. The second τ ∗ Seq satisfies both, the separating properties and sequentiality, eventhough it has a drawback from the practical viewpoint, namely, one does not have a priori anexplicit expression for L τ ∗ in terms of L . Finally, the third τ L ∗ may depend on the choice of L for the topology τ , but it coincides with τ ∗ Seq when L is of first order (i.e. L = L τ ). So, oneobtains the explicit limit operator L τ ∗ = L ∗ . 12 eneral topology CHOICE OF D ⊂ X Minim. D-separatingTheorem 3.3topology τ ∗ Minim. sequentiallyD-separatingtopol. τ ∗ Seq = τ Lτ ∗ τ L ∗ with L ∗ defined from L (Maximal limit oper. when τ is sequential τ = τ L for somelimit operator L Coincide if L of first order( L = L τ ), Theorem 3.9 Defn. inform. (8)
Theorem 3.5 τ –sense of Prop. 3.7) Figure 1: Summary of the constructions and results of Section 3.Although the first order restriction for L is not fundamental in this context (see the previousremark), we will consider also non-necessarily first order limit operator L , as this might be thecase for the natural operator for the chronological topology. So, the next proposition gives asmall extension of the previous theorem, to be used later.In what follows, τ will be a sequential topology derived from some limit operator L , i.e. τ = τ L . Lemma 3.11 If p ∈ X \ D satisfies p ∈ L ∗ τ ( σ ) for some sequence σ ⊂ X , then L i ( σ ) = ( L ∗ ) i ( σ ) for any ordinal i .Proof: The inclusion to the left is trivial, so we will focus on the inclusion to the right. We willproceed by transfinite induction. For i = 1, the equality follows from the definition of L ∗ andthe fact that no subsequence κ of σ satisfies d ∈ L ( κ ) ∩ D ; in fact, otherwise, d ∈ L τ ( κ ), andthus, L ∗ τ ( σ ) ⊂ D , which implies p (cid:54)∈ L ∗ τ ( σ ), a contradiction. So, let us assume that the result isvalid for any j < i , and let us prove it for i . Recall first that L ( κ ) ∩ D = ∅ for any subsequence κ of { p n } ⊂ ∪ j
Let L be a limit operator of k -th order (for some ordinal k ) which is also offirst order on D . If L ∗ is of first order then L ∗ = L ∗ τ (where L ∗ τ := ( L τ ) ∗ ).Proof: Since L ( σ ) ⊂ L τ ( σ ) for all σ ⊂ X , and L is of first order on D , we directly have L ∗ ( σ ) ⊂ L ∗ τ ( σ ) . For the inclusion to the left, consider a point p ∈ L ∗ τ ( σ ). If p ∈ D then p ∈ L ∗ τ ( σ ) ⇒ p ∈ L τ ( σ ) ⇒ p ∈ L ( σ ) ⇒ p ∈ L ∗ ( σ ) ,
13s required (we have used (8) for the first and third implications, and the first order characterof L on the open set D for the second implication). If, otherwise p ∈ X \ D , from Lemma 3.11, L i ( σ ) = ( L ∗ ) i ( σ ) for any ordinal i . Moreover, as L ∗ is of first order, L ∗ ( σ ) = ( L ∗ ) i ( σ ) for anyordinal i (recall Corollary 2.10). Therefore, taking into account that L is a limit operator of k -thorder, we have p ∈ L ∗ τ ( σ ) ⊂ L τ ( σ ) = L k ( σ ) = ( L ∗ ) k ( σ ) = L ∗ ( σ ) . In conclusion, in both cases we deduce that p ∈ L ∗ ( σ ), as desired. T -separability and c-boundary Let (
M, g ) be a spacetime, i.e. a connected time-oriented Lorentzian manifold. There are severaltypes of boundaries in Relativity applicable to (some classes of) spacetimes, yielding a completionof the spacetime M = M ∪ ∂M . Among them, one has Geroch geodesic boundary (g-boundary)[13], Schmidt bundle boundary (b-boundary) [25, 26], Scott and Szekeres abstract boundary (a-boundary) [24], Penrose conformal boundary [21, 28] and Geroch, Kronheimer and Penrose causalboundary (c-boundary).The g-boundary was defined by using classes of incomplete geodesics in ( M, g ). For the b-boundary, one defines a certain positive definite metric on the bundle of linear frames LM of M ,takes the Cauchy completion of LM and induces then a boundary for ( M, g ). Both constructionssatisfy the following a priori desirable properties pointed out by Geroch, Liang and Wald [15]:(i) every incomplete geodesic in the original spacetime terminates at a point, and(ii) they are geodesically continuous, in a sense rigorously defined in [15].However, Geroch. et al. found a drawback for any boundary satisfying (i) and (ii): in a simpleinsightful example of (stably causal) spacetime, these two conditions implied the existence of apoint r in the spacetime and a point s in the boundary non- T -separated (see Fig. 2); the authorsproposed even a refined version of the example that was flat. The topology in the exampleswas always sequential and, moreover, there was a natural intuitive sense of convergence in theformulation of (ii). So, this drawback was regarded as a reason to reject a priori the topologiessatisfying them.Now, recall that our previous results can be applied. More precisely, putting X equal to theunion of M and its g- or b- (or other) boundary ∂M , and D = M , one can modify slightly theoriginal topology τ by taking the minimally D -separating one τ ∗ provided by Theorem 3.3 (orthe sequentially minimally D -separating one τ ∗ Seq in Theorem 3.5, if sequentially is also requiredto be preserved). This topology does not present the commented drawback and can be developedfurther. Of course, this modification of the topology will not solve magically all the problems ofthese boundaries, but it suggests possible improvements and opens the opportunity to re-studythem (see, for instance, [23]).Let us explain this briefly for the g-boundary (even though the b-boundary is more elegant andappealing mathematically [2, 17], it has other type of drawbacks from the physical viewpoint,see [5, 22]). As a previous question, one has to ensure that, if the spacetime is dense for thetopology τ of the original completion, it is also dense for the modified one τ ∗ . For this purpose,the following straightforward proposition is clearly applicable to the g-boundary, as well as tothe b-, c- and conformal boundaries: Proposition 4.1
Let ( X, τ ) be a topological space and D ⊂ X a Hausdorff locally compact opensubset of X . Assume that for any x ∈ X \ D there exists some sequence σ ⊂ D such that ∈ L τ ( σ ) . Then, D is dense in X for τ . If, in addition, such a σ can be chosen such that itsatisfies L τ ( σ ) = L ∗ τ ( σ ) , then D is also dense for τ ∗ Seq . Now, our modified topology τ ∗ for the g-boundary must violate one of the two properties (i) or(ii) above. Clearly, (i) will not be violated because of the minimality of our modification (recallthe previous proposition). So, the key is the meaning of the hypothesis geodesically continuous in (ii). Geroch, Liang and Wald introduced a natural definition of this concept in terms of theexponential map. In addition, they gave a clever counterexample ( M, g ) to the property of T -separability, by exploiting the following property (see Figure 2 as well as [15]): removing a point s in Lorentz-Minkowski L , they found a metric g (conformal to the usual one) and two points p, r ∈ M joined by a sequence of (future-directed) timelike geodesics γ i of length equal to 1 suchthat; (a) their initial velocities ξ i = γ (cid:48) i (0) converge to a timelike vector ξ , and (b) the geodesic γ with ξ = γ (cid:48) (0) satisfy that lim t (cid:37) γ ( t ) is the removed point s . Their conclusion was that s should be identified with a point of the g-boundary, and any neighborhood of this point contains r . Our modified topology τ ∗ separates r and s , and seems to give a reasonble behavior for thetopology in this particular example. Nevertheless, this does not simply mean that τ ∗ must bethe right topology for the g-construction (notice that Geroch had also suggested in his originalarticle the possibility of a modification of the g-topology, see the footnotes 10 and 14 in [13]).A closer look to Geroch et al.’s counterexample shows that they took advantage of the lack ofcompactness of J + ( p ) ∩ J − ( r ) as well as the lack of good convergence properties in the closureof a convex set. So, a first test for a redefinition might be carried out by restricting the class ofspacetimes (for example, starting at globally hyperbolic ones). We leave this question as openfor possible future studies.The a-boundary, conformal boundary and c-boundary are not affected a priori by previousobjection, as they are not formulated in terms of geodesics. The a-boundary [24] is defined interms of open embeddings and has been developed systematically at the level of the set of idealpoints. At the topological level, Barry and Scott have introduced recently two topologies for thea-boundary, the so-called attached and strongly attached point topologies (see [3, 4] for details). Byconstruction, these topologies are not affected by the problem of lack of Hausdorffness betweenmanifold and boundary points, and they have other desirable properties (in fact, their secondproposal improves some properties of the first one). The conformal boundary is defined in termsof open conformal embeddings in a bigger (and Hausdorff) spacetime; so, it is not affected by theseproblems of separability. Nevertheless, harder problems appear in order to ensure the existenceof such conformal embeddings and, in this case, the uniqueness of the so-obtained conformalboundary. The c-boundary appears as a natural alternative to the conformal boundary. In fact,the c-completion of spacetimes is a conformally invariant and systematic construction, applicableto any strongly causal spacetime. The c-completion of a strongly causal spacetime is constructed by adding ideal points to thespacetime in such a way that any timelike curve in the original spacetime acquires some end-point in the new space. In the original article by Geroch, Kronheimer and Penrose [14], somedoubts on its topology were pointed out. In fact, the original definition has suffered quite a fewmodifications. We will consider here the recent redefinition in [9] (which includes an approach tothe topology of partial boundaries by Harris [18, 19]), and refer also there for extensive bibliog-raphy on the topic (see also [20, 10, 11] and references therein). This c-completion of a spacetimeis always T , and there are situations where the possible non- T separation of two points of theboundary appears as natural. However, there exist still examples where a point of M and one15 γ γ U U ps r ξ ξ ξ Figure 2: In their example, Geroch, Liang and Wald considered R \ { s } endowed with a metricΩ · η , where η is the Minkowski metric and Ω is a conformal factor. The conformalfactor was defined in such a way that all γ i are geodesics and Ω ≡ U i as well as on each curve γ i inside U i (see [15] for details).16f its c-boundary ∂M are not T -related (see Example 6.1). As in the case of the g- and b-boundaries, one can get rid of this inconvenience by using the minimal modification of the orig-inal topology in Theorems 3.3 and 3.5. In the remainder of Section 4, we will develop in detailthis modification, in order to check that all the other desirable properties of the c-boundary arepreserved by this minimal modification.First, we will introduce some basic notions. A non-empty subset P ⊂ M is called a past set if it coincides with its past; i.e., P = I − ( P ) := { p ∈ M : p (cid:28) q for some q ∈ P } . The commonpast of S ⊂ M is defined by ↓ S := I − ( { p ∈ M : p (cid:28) q ∀ q ∈ S } ). In particular, thepast and common past sets are open. A past set that cannot be written as the union of twoproper past sets is called indecomposable past set, IP . An IP which does coincide with the pastof some point of the spacetime P = I − ( p ), p ∈ M is called proper indecomposable past set , PIP .Otherwise, P = I − ( γ ) for some inextendible future-directed timelike curve γ , and it is called terminal indecomposable past set , TIP . The dual notions of future set , common future , IF , PIF and
TIF , are defined just by interchanging the roles of past and future in previous definitions.To construct the future and past c-completion , first we have to identify each event (point) p ∈ M with its PIP, I − ( p ), and PIF, I + ( p ). This is possible in any distinguishing spacetime, that is,a spacetime which satisfies that two distinct events p, q have distinct chronological futures andpasts ( p (cid:54) = q ⇒ I ± ( p ) (cid:54) = I ± ( q )). In order to obtain consistent topologies in the c-completions,we will focus on a somewhat more restrictive class of spacetimes, the strongly causal ones . Theseare characterized by the fact that the PIPs and PIFs constitute a sub-basis for the topology ofthe manifold M .Now, every event p ∈ M can be identified with its PIP, I − ( p ). So, the future c-boundary ˆ ∂M of M is defined as the set of all the TIPs in M , and the future c-completion ˆ M becomes the setof all the IPs: M ≡ PIPs , ˆ ∂M ≡ TIPs , ˆ M ≡ IPs . Analogously, each p ∈ M can be identified with its PIF, I + ( p ). The past c-boundary ˇ ∂M of M is defined as the set of all the TIFs in M , and the past c-completion ˇ M is the set of all the IFs: M ≡ PIFs , ˇ ∂M ≡ TIFs , ˇ M ≡ IFs . For the (total) c-boundary, the so-called S-relation comes into play [27]. Denote ˆ M ∅ = ˆ M ∪{∅} (resp. ˇ M ∅ = ˇ M ∪ {∅} ). The S-relation ∼ S is defined in ˆ M ∅ × ˇ M ∅ as follows. First, in the case( P, F ) ∈ ˆ M × ˇ M , P ∼ S F ⇐⇒ (cid:26) P is included and is a maximal IP into ↓ FF is included and is a maximal IF into ↑ P. (12)By maximal we mean that no other P (cid:48) ∈ ˆ M (resp. F (cid:48) ∈ ˇ M ) satisfies the stated property andincludes strictly P (resp. F ). Recall that, as proved by Szabados [27], I − ( p ) ∼ S I + ( p ) for all p ∈ M , and these are the unique S-relations (according to our definition (12)) involving properindecomposable sets. Now, in the case ( P, F ) ∈ ˆ M ∅ × ˇ M ∅ \ { ( ∅ , ∅ ) } , we also put P ∼ S ∅ , (resp. ∅ ∼ S F )if P (resp. F ) is a (non-empty, necessarily terminal) indecomposable past (resp. future) set thatis not S-related by (12) to any other indecomposable set; notice that ∅ is never S-related to itself.Now, we can introduce the notion of c-completion, according to [9]: Definition 4.2
Let ( M, g ) be a strongly causal spacetime. The c-completion of ( M, g ) is definedas follows. As a point set: the c-completion M is formed by the pairs of TIPs and TIFs which areS-related, that is, M := { ( P, F ) ∈ ˆ M ∅ × ˇ M ∅ : P ∼ S F } . Every point p ∈ M of the manifold will be identified with its corresponding pair ( I − ( p ) , I + ( p )) ,so M will be considered a subset of M and, thus, the c-boundary is defined as ∂M = M \ M . • As a chronological set: two pairs ( P, F ) , ( P (cid:48) , F (cid:48) ) ∈ M are chronologically related, denoted ( P, F ) (cid:28) ( P (cid:48) , F (cid:48) ) , if F ∩ P (cid:48) (cid:54) = ∅ . • Topologically: M is endowed with the chronological topology τ chr , i.e., the sequential topol-ogy associated to the following limit operator (recall Definition 2.1): L chr ( σ ) := (cid:26) ( P, F ) ∈ M : P ∈ ˆ L ( { P n } ) if P (cid:54) = ∅ F ∈ ˇ L ( { F n } ) if F (cid:54) = ∅ (cid:27) for any σ = { ( P n , F n ) } ⊂ M ,where P ∈ ˆ L ( { P n } ) ⇐⇒ P ⊂ LI( { P n } ) and it is maximal in LS( { P n } ) F ∈ ˇ L ( { F n } ) ⇐⇒ F ⊂ LI( { F n } ) and it is maximal in LS( { F n } ) . The main properties of this choice of c-completion (
M , (cid:28) , τ chr ) are summarized in the followingresult (see [9, Theorem 3.27]):
Theorem 4.3
The c-completion ( M , (cid:28) , τ chr ) in Defn. 4.2 satisfies the following properties:(i) Any future-directed (resp. past-directed) inextendible timelike curve in M (namely, γ :[ a, b ) → M ) has an endpoint in ∂M .(ii) The inclusion i : M (cid:44) → M is a topological embedding and i ( M ) is dense in M .(iii) The c-boundary ∂M is closed in M .(iv) I ± (( P, F )) is open for any ( P, F ) ∈ M .(v) The topology τ chr is T . Given the limit operator L chr and its associated chronological topology τ chr := τ L chr on M , weput D = M and consider the minimally M -separating topology τ ∗ chr , the minimally sequentially M -separating topology ( τ ∗ chr ) Seq , the operator L ∗ chr and its associated topology τ L ∗ chr . In thebeginning of Section 3, we imposed a “minimality” condition for our refinements in order toensure that both, the original topology and the refined ones shared as many properties as possible(except the new properties). Therefore, it is expected that the refinements defined along sucha section (summarized in Remark 3.10 and Figure 1) will also satisfy the properties included inTheorem 4.3. In fact, we can prove the following result: Theorem 4.4
The topological spaces ( M , τ ∗ chr ) , ( M , ( τ ∗ chr ) Seq ) and ( M , τ L ∗ chr ) endowed with thechronological relation (cid:28) satisfy all the assertions (i)–(v) in Theorem 4.3. Moreover, the pointson M are T -separated from the points on ∂M with the topologies τ ∗ chr and ( τ ∗ chr ) Seq , and thelatter is equal to τ L ∗ chr if L chr is of first order. roof. Assertions (iii), (iv) and (v) are straightforward as all topologies are finer than thechronological one, and this one satisfies them (by Theorem 4.3). Next, recall that, for any future(resp. past) inextendible timelike curve γ : [ a, b ) → M and any sequence t n (cid:37) b , the followingequation L ( { γ ( t n ) } ) = { ( P, F ) ∈ ∂M : P = I − ( γ ) } (resp. L ( { γ ( t n ) } ) = { ( P, F ) ∈ ∂M : F = I + ( γ ) } )is true for both, L chr and L ∗ chr . So, assertion (i) is implicitly proved in [9, Theorem 3.27], whilethe second part of (ii) is a consequence of the fact that no sequence { γ ( t n ) } with t n (cid:37) b convergesto a point in M . On the other hand, the first assertion in (ii) follows from the first assertion ofTheorem 4.3 (ii) and the fact that L ∗ chr | M = L chr | M (recall (8), i.e., formula (14) below andProposition 2.7). Finally, the last assertion is obtained from Theorem 3.9. Remark 4.5 (1) The topology τ ∗ chr may not be Hausdorff, as two points at the boundary ∂M may be non-T -separated. However, such type of examples are natural in different situations.Namely, this happens if one removes a half time axis of Lorentz-Minkowski L (as the removedorigin would yield naturally two boundary points), or in more refined examples such as the“grapefruit on a stick” in [7] or the two chimneys construction in [11, Appendix]. In theseexamples, the ideal points which are non-T -separated for τ chr remain non-T -separated for τ ∗ chr —but this can be regarded as harmless.(2) Conditions (i)–(v) of both, Theorem 4.3 and Theorem 4.4, remain true with independenceof the fact that L chr or L ∗ chr may or not be of first order. The first order condition is onlynecessary to ensure that τ L ∗ chr satisfies (A Sep ). A strong support for the choice of the explained definition of c-completion in [9], is that such achoice follows from a set of minimal hypotheses, which catch the intuitive requirements that ac-boundary must fulfill. But, as pointed out in that reference, one could add more hypotheses iffurther properties were required for that boundary. Let us revisit first the original hypotheses,and then, let us add the minimal separability of the boundary as one of these hypotheses.
The admissibility conditions for the c-boundary in [9] provide the point set and chronologicalstructures previously explained in subsection 4.2. Let us remind those for the topology.
Definition 5.1
Consider a strongly causal spacetime M and its c-completion M regarded onlyas a point set and a chronological set (see Defn. 4.2). The following hypotheses are conditionsof admissibility for a topology τ on M :(A1) For all ( P, F ) ∈ M , the sets I ± (( P, F )) are open.(A2) The limits for τ are compatible with the empty set, i.e., if { ( P n , F n ) } n → ( P, ∅ ) (resp. ( ∅ , F ) ) and there exists ( P (cid:48) , F (cid:48) ) ∈ M such that P ⊂ P (cid:48) ⊂ LI( { P n } ) (resp. F ⊂ F (cid:48) ⊂ LI( { F n } ) ), then ( P (cid:48) , F (cid:48) ) = ( P, ∅ ) (resp. ( P (cid:48) , F (cid:48) ) = ( ∅ , F ) ).( A Min ) τ is minimally fine among the topologies satisfying previous conditions, i.e., no other topol-ogy satisfying the conditions (A1) and (A2) is strictly coarser than τ . A Seq
Min ) τ is minimally fine among the sequential topologies satisfying previous conditions (A1) and(A2), i.e., τ is sequential and no other sequential topology satisfying conditions (A1) and(A2) is strictly coarser than τ .A topology τ satisfying the conditions (A1), (A2) and ( A Min ) will be called generally admissible and one satisfying (A1), (A2) and ( A Seq
Min ) will be sequentially admissible or simply admissible .Accordingly, the c-completion M endowed with such a topology will be called generally admissible or admissible . Remark 5.2
Here we have reserved the term “admissible topology” for the plain notion definedabove, which agrees with [9] and subsequent references. When, in the spirit of [9, Sect. 3.1.3],further hypotheses are imposed, we will use a different name (as in Definition 5.6 below), in orderto avoid confusions.Let us discuss briefly the conditions in Definition 5.1 (see [9] for further details). Condition (A1)determines partially the convergence of sequences. In fact, for any topology τ satisfying (A1):( P n , F n ) → ( P, F ) ⇒ P ⊂ LI( { P n } ) , F ⊂ LI( { F n } ) . Moreover, if, in addition, P (cid:54) = ∅ (cid:54) = F , we have the following implication in terms of the operator L chr of the chr-chronology:( P n , F n ) → ( P, F ) ⇒ ( P, F ) ∈ L chr ( { ( P n , F n ) } ) . (13)(see [9, Lemma 3.15] for details). Condition (A2) is a compatibility requirement for the con-vergence of sequences when a component of the limit pair is empty. In particular, one canextend formula (13) to any ( P, F ) ∈ M , that is, if a topology τ satisfies (A1) and (A2) and( P n , F n ) → ( P, F ) one has:( P n , F n ) → ( P, ∅ ) ⇒ ( P, ∅ ) ∈ L chr ( { ( P n , F n ) } )(resp. ( P n , F n ) → ( ∅ , F ) ⇒ ( ∅ , F ) ∈ L chr ( { ( P n , F n ) } )) . (see [9, Prop. 3.20] for details). Summing up, the following property is obtained for referencing: Lemma 5.3 If σ → ( P, F ) with any topology τ satisfying (A1), (A2) then ( P, F ) ∈ L chr ( σ ) . Finally, conditions (A
Min ) or (A
SeqMin ) are minimality conditions which will allow to speak onuniqueness and guarantee that no spurious sets are included in the topology.From the viewpoint of “first principles”, the following result, which clarifies [9, Theorem 3.22],justifies the choice of the chronological topology for the c-completion (at least when L chr is offirst order). Theorem 5.4
Let M be the c-completion of a strongly causal spacetime M regarded as a pointset and a chronological set. If the limit operator L chr is of first order, then the chr-topologyis the unique (sequentially) admissible topology on M . Moreover, in this case, L chr is also theassociated limit operator of any generally admissible topology τ ⊂ τ chr .Proof. The first assertion is obtained in [9, Theorem 3.22], but we can sketch now a proof by usingour previous results as follows. First, a straightforward computation shows that L chr satisfies(A1) and (A2). By Lemma 5.3, the limit operator L associated to any sequential topologyobeying (A1) and (A2) must satisfy L ⊂ L chr , and the minimality assumption (A SeqMin ) implies L = L chr . 20or the last assertion, assume that there exists a generally admissible topology τ ⊂ τ chr anddenote by L τ its associated limit operator. On the one hand, from Proposition 2.5 (b), wededuce that τ Seq ⊂ τ L chr and, as L τ is of first order, Lemma 2.4 ensures that L chr ( σ ) ⊂ L τ ( σ )for all sequence σ ⊂ M . On the other hand, recall that the limit operator L τ is associated to atopology τ fulfilling the admissibility conditions, and so, it must also obey L τ ( σ ) ⊂ L chr ( σ ) forall sequence by Lemma 5.3. In conclusion, L τ = L chr . Remark 5.5
As the operator L chr is not of first order only in very artificial cases (see Example6.2), this theorem (plus the properties of Theorem 4.3) gives a strong support for the usageof the chr-topology, at least if one does not care on the T -separability of the boundary. Infact, if a redefinition of the c-boundary topology preserved the expected good properties, then itwould agree with the chr-topology whenever L chr is of first order. Moreover, in this case the lastassertion of previous theorem ensures that, whenever generally admissible topologies exist, theyare minimally fine among the topologies with L chr as an associated limit. When L chr is not offirst order then the chronological topology is not admissible (this can be checked directly fromLemma 5.3). However, the chronological topology would still make sense and we will exploreeven this case. D -separability as an admissibility condition As shown in Section 3, the T -separation of points of M and ∂M can be always obtained asa requirement a posteriori . However, one can think that such a property should be includedas one of the a priori properties of the c-boundary, at the same level of the other admissibilityconditions. Next, this approach is developed.Recall first that the limit operator of the searched ( T -separating) topology τ must satisfy alsothe admissibility conditions (A1) and (A2) and, so, Lemma 5.3 implies L τ ⊂ L chr . We focuson the case of sequential topologies because, on the one hand, the topology of the c-boundaryis studied by considering the convergence of some sequences and, on the other, results suchas Theorem 5.4 show that they are specially interesting (of course, this can be extended tonon-sequential ones, as we have done till now). Definition 5.6
A sequential topology τ is T -admissible for the c-completion M if it satisfiesproperties (A1) and (A2) of Definition 5.1, together with (A Sep ) in Definition 3.1 and the fol-lowing one:( ˜ A SeqMin ) τ is minimally fine among the sequential topologies satisfying previous conditions, i.e., noother sequential topology satisfying (A1), (A2) and (A Sep ) is strictly coarser than τ . Taking into account Proposition 3.7, we consider the operator L ∗ chr ( σ ) := (cid:26) L chr ( σ ) ∩ M if ∃ κ ⊂ σ and p ∈ M such that p ∈ L chr ( κ ) L chr ( σ ) otherwise (14)consistently with (8). If we apply Theorem 3.9 (ii) to ( M , τ chr ) and D = M , we deduce directly: Corollary 5.7
Let M be the c-completion of a strongly causal spacetime M regarded as a pointset and a chronological set. If the limit operator L chr is of first order, then the operator L ∗ chr isalso of first order, and its associated topology τ L ∗ chr is equal to ( τ ∗ chr ) Seq , i.e., τ L ∗ chr is the uniquesequentially minimally D -separating topology for D = M . τ L ∗ chr is the suitable topology for therequirements in Definition 5.6 (in the same sense that Theorem 5.4 proved that τ L chr was thesuitable topology for Definition 5.1) by imposing directly the first order character to L ∗ chr . Theorem 5.8
Let M be the c-completion of a strongly causal spacetime M regarded as a pointset and a chronological set. If L ∗ chr is of first order on M , then the topology τ L ∗ chr satisfies (A1),(A2) and (A Sep ). If, in addition, L chr is of k -th order for some ordinal k (in particular, bothhypotheses hold when L chr is of first order), then τ L ∗ chr also satisfies ( ˜ A SeqMin ), i.e., τ L ∗ chr is theunique T -admissible sequential topology of M .Proof. Let us begin with the first assertion, that is, let us show that τ L ∗ chr satisfies (A1), (A2)and (A Sep ) if L ∗ chr is of first order. Since L ∗ chr ⊂ L chr , thus τ chr ⊂ τ L ∗ chr (see Lemma 2.4),and τ chr satisfies (A1), we deduce that τ ∗ chr also satisfies (A1). Moreover, from Proposition3.7 and Lemma 3.6, we know that τ L ∗ chr satisfies (A Sep ). In order to prove (A2), assume that σ = { ( P n , F n ) } → ( P, ∅ ), P (cid:54) = ∅ , with τ ∗ chr (the case P = ∅ (cid:54) = F is analogous) and assumethe existence of ( P (cid:48) , F (cid:48) ) ∈ M such that P ⊂ P (cid:48) ⊂ LI( { P n } ). As L ∗ chr is of first order on M ,( P, ∅ ) ∈ L ∗ chr ( σ ). Taking into account that ( P, ∅ ) ∈ ∂M , necessarily L ∗ chr ( σ ) = L chr ( σ ), andthus, ( P, ∅ ) ∈ L chr ( σ ). In particular, P is a maximal IP in LS( { P n } ). Therefore, as ( P (cid:48) , F (cid:48) ) ∈ M satisfies P ⊂ P (cid:48) ⊂ LI( { P n } ), necessarily P = P (cid:48) and F (cid:48) = ∅ , as required.For the second assertion, assume additionally that L chr is of k -th order and let us show that τ L ∗ chr also satisfies ( ˜A Seq
Min ). In fact, we are going to prove something even stronger, concretely, thatany sequential topology τ L satisfying (A1), (A2) and (A Sep ), must obey τ L ∗ chr ⊂ τ L . On the onehand, as L chr is of k -th order, Proposition 3.12 ensures that L ∗ chr = L ∗ τ chr , and so, from Theorem3.9 (ii), we deduce that τ L ∗ chr is the unique sequentially minimally D -separating topology. Onthe other hand, as τ L satisfies (A1) and (A2), Lemma 5.3 ensures that L ( σ ) ⊂ L chr ( σ ) forall sequence σ and so, τ chr ⊂ τ L , i.e., τ L satisfies (A Fin ). Taking into account that τ L alsosatisfies (A Sep ) and the definition of sequentially minimally D -separating topology, we deducethat τ L ∗ chr ⊂ τ L , as desired.Figure 3 summarizes the results of this section, for the convenience of the reader. Remark 5.9
As the hypotheses of Theorem 5.8 are less restrictive than the first order propertyfor L chr , its conclusion is sharper than Corollary 5.7 (see Example 6.2). But, beyond this subtlety,either Theorem 5.8 or Corollary 5.7 show that no matter if the T -separability is imposed a priori( T -admissible) or a posteriori ( D -separating).Summing up, the question of T -separability can be circumvented for the c-boundary. From afundamental viewpoint, one can work with the T -separating topology τ L ∗ chr , as the admissibilityproperties of this topology yield more accurate consequences than those for τ L chr (Corollary 5.7,Theorem 5.8). In any case, as pointed out in Remark 5.5, the cases where the topologies τ chr or τ L ∗ chr satisfy the conditions of admissibility are so general that one would not be especially worriedwith the cases where they differ. And, from the practical viewpoint, this way of proceeding isequivalent to consider the chr-completion (as defined in [9]) and impose the T -separation of M and ∂M , as one can do in the general framework of Section 3. Nevertheless, if further studiessuggested that more separability properties must be required for the c-boundary, one couldremake our previous process by including such properties, in the spirit of [9]. This section is devoted to present some examples in order to illustrate some of the assertionsappeared in previous sections. The first two examples were already studied in [8] and [1], resp.,22 dmisibilityChronologicaltopology τ chr Chronological limitoperator L chr (A1), (A2) and ( ˜A Seq
Min )Refining the topology with D = M When L chr is of first order Formula (14)Definition 4.2When L chr of first order L ∗ chr → derived top. τ L ∗ chr (Theorem 5.4) T -admissibility:(A1), (A2), (A Sep )and ( ˜A
Seq
Min )(Theorem 3.9) When L chr is of first order (Corollary 5.7)or more generally L ∗ chr of first order and L chr of k -th order (Theorem 5.8) τ ∗ chr → limit op. L τ ∗ chr → derived topology ( τ ∗ chr ) Seq
Theorem 3.3 (recall (2))
Figure 3: This figure summarizes the main results in Section 5.Recall that Section 3 provides two different ways to refine the topology τ chr in order toobtain the separability condition on D = M : (a) focusing on the chronological topologyitself (following the dotted arrow) or (b) considering the limit operator L chr (followingthe dashed arrow). When L ∗ chr is of first order the procedure (b) is equivalent to include T -separability as an admissibility condition for the c-boundary (Theorem 5.8). In anycase, when L chr is of first order (which is slightly more restrictive than L ∗ chr of firstorder), the procedures (a) and (b) are equivalent (Theorem 3.9).23 xFPy p n p Figure 4: Minkowski spacetime L with the dashed regions (two discs of radius 2 centered at thepoints ( − , ,
1) and (1 , , − Example 6.1
Let us begin by showing the possibility for the c-boundary of the setting thatmotivates this paper, that is, the existence of a c-completion M where two points, one on themanifold and other on the boundary, are not T -related. For this, we will just recall the twoexamples introduced in [8, Section 2.3]. Such examples show not only that the c-completion maypresent separability problems, but also that the involved boundary points can be represented byany type of pairs ( P, F ) (i.e., with both or just one non-empty component).On the first example (represented in Figure 4), the non-empty sets P and F are S-related, andso, they determine a point at the boundary. Moreover, from the definition of the chronologicallimit (recall Def. 4.2), we have that the sequence { p n } converges to both, the manifold point p ∈ M and the boundary point ( P, F ) ∈ ∂M .On the second example (Figures 5 and 6), there are two boundary points attached to q . Infact, the set P (which includes P (cid:48) but it is indecomposable) is S-related with F while the set P (cid:48) is S-related with the empty set, obtaining the boundary points ( P, F ) , ( P (cid:48) , ∅ ) ∈ ∂M . Here, thesequence { p n } converges again to both, a point in the manifold p ∈ M , and a boundary point having one empty component ( P (cid:48) , ∅ ) ∈ ∂M . Example 6.2
We are going to show that the limit operator L chr of the completion of a spacetimeis not necessarily of first order (suggesting a procedure for the construction of examples of k -thorder for any k ∈ N ∪ {ℵ } ). To this aim, we explain first a key example which is described withfurther details in [1, Chapter 4].The spacetime M consists of L with the following subsets removed (see Figures 7, 8): thecausal future J + (0 , , { Π ∞ n } n ≥ , where Π ∞ n = J − (0 , , ∩ Π n and Π n is the plane t − /n = 0, the semi-discs { D n } n ≥ ∪ D ∞ and the sheets { Π ln } l,n ≥ (obtained from { Π ∞ n } n by aconvenient “contraction and translation”). The spacetime M presents a number of distinguishedindecomposable sets, which are also depicted in the figures, say: P ∞ , P (cid:48)∞ , F = I + ( p ) and24 xyp q p n FP’ P Π n D Figure 5: Representation of the spacetime M , obtained by removing from L the dashed regions,which consist of the unitary disc D and the sheets Π n (see Figure 6 for a lateral viewof the sheets, which allows to distinguish between P and P (cid:48) ). yt L n o P (cid:48) Pγ (cid:48) γ Figure 6: Slice of M at x = x , as seen from the “eye” in previous figure. The segments L n = π n ∩ { x = x } for some x ∈ R are represented. The curves γ and γ (cid:48) which defines P and P (cid:48) as IPs are here also depicted. In fact, P = I − ( γ ) includes the region betweenthe two half lines at ±
45 degrees with respect to the axis, while P (cid:48) = I − ( γ (cid:48) ) is onlythe subregion striped with thin lines at ±
45 degrees.25 = ( , , ) p p p p p ∞ = (0 , , D D D D D ∞ P ∞ P (cid:48)∞ P P F Figure 7: Representation of the spacetime M , obtained by removing from L the grey regions.The figure also shows the distinguished IPs, P ∞ , P (cid:48)∞ and { P n } n ≥ (see Figure 8).Observe that every P n and P (cid:48) n consists of a “contraction plus a translation” of P ∞ and P (cid:48)∞ resp. P n , P (cid:48) n with n ≥
1; notice that each ( P n , F ) (rather than p n ) as well as ( P ∞ , F ) and ( P (cid:48)∞ , ∅ ) arec-boundary points.Consider the sequence σ = { x n } ⊂ M , with x n = (1 / , /n, /
2) for all n . Then, rea-soning as in Example 6.1, L chr ( σ ) contains { ( P n , F ) } n ≥ and ( P ∞ , F ). Moreover, ( P (cid:48)∞ , ∅ ) ∈ L chr ( { ( P n , F ) } ). So, σ converges to ( P (cid:48)∞ , ∅ ) with the chronological topology, as ( P (cid:48)∞ , ∅ ) ∈ L chr ( σ )(recall (5) and Proposition 2.8). Note, however, that P (cid:48)∞ (cid:40) P ∞ ⊂ LI ( { I − ( x n ) } )). Hence, P (cid:48)∞ is not a maximal IP in LS ( { I − ( x n ) } ), and thus, ( P (cid:48)∞ , ∅ ) (cid:54)∈ L chr ( σ ). In conclusion, L chr isnot of first order. However, it seems that all the possible limits are contained in L chr (andmaking a straightforward modification of the example it would be contained in some L kchr ), soit is conceivable the existence of examples where L chr is a k -th order limit operator for any k ∈ N ∪ {ℵ } . Example 6.3
Here, we present an example clarifying our definition of condition (A
Sep ). Con-cretely, we will justify the requirement that the open set U belongs to the original topology τ M of the manifold M and not to the topology to be obtained. Basically, the reason of our choiceis that the alternative topology might not preserve the original topology of the manifold M , asthe following example shows.Consider the spacetime M depicted in Figure 5. It is not difficult to check that the chronological26 P (cid:48) Π n D D P P (cid:48) Π n P ∞ P (cid:48)∞ D ∞ Π ∞ n Figure 8: Section of M as seen from the “eye” in previous figure. Note that, between twosemi-discs D i and D i +1 , we have the same effect as in Example 6.1 (compare withFigures 5 and 6). Moreover, for all n , P ∞ (cid:54)⊂ P n (in fact, P m (cid:54)⊂ P n for m > n ), but P (cid:48)∞ ⊂ P n . This implies P (cid:48)∞ ⊂ LI ( { P n } ) and P (cid:48)∞ is maximal IP in LS ( { P n } ), that is,( P (cid:48)∞ , ∅ ) ∈ L chr ( { ( P n , F ) } ). 27imit operator L chr is of first order. Let us define a new limit operator L ∗∗ chr as follows (comparewith (8)): L ∗∗ chr ( σ ) := (cid:26) L chr ( σ ) ∩ ∂M if ∃ p ∈ M and ( P, F ) ∈ ∂M such that p, ( P, F ) ∈ L chr ( σ ) L chr ( σ ) otherwise,and denote by τ ∗∗ chr the topology associated to this limit operator. Since L chr is of first order, itfollows, by the same arguments as in Section 3.3, that L ∗∗ chr is a first order limit operator (recallRemark 5.9). Moreover, its associated topology τ ∗∗ chr is finer than τ chr , T -separates the pointsof the boundary from the points of the manifold, and it is minimally finer among the sequentialones satisfying such properties. However, the sequence σ = { p n } in Figure 5 does not convergeto p with τ ∗∗ chr , since it satisfies p, ( P (cid:48) , ∅ ) ∈ L chr ( σ ) and so, p / ∈ L ∗∗ chr ( σ ). Taking into accountthat σ clearly converges to p with the manifold topology, we deduce that τ ∗∗ chr does not preservesthe manifold topology on M . Acknowledgments
The authors are partially supported by the Spanish Grants MTM2013-47828-C2-1-P and MTM2013-47828-C2-2-P (MINECO and FEDER funds). The second-named author is also supported byFAPESP (Funda¸c˜ao de Amparo ´a Pesquisa do Estado de S˜ao Paulo, Brazil) Process 2012/11950-7.
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