The Attached Point Topology of the Abstract Boundary For Space-Time
aa r X i v : . [ g r- q c ] A ug Published: Richard A Barry and Susan M Scott 2011 Class. QuantumGrav. 28(16) 165003 doi: 10.1088/0264-9381/28/16/165003
The Attached Point Topology of the Abstract Boundary ForSpace-Time
Richard A Barry and Susan M Scott
Centre for Gravitational Physics, College of Physical and MathematicalSciences, The Australian National University, Canberra ACT 0200,Australia [email protected], [email protected]
Abstract
Singularities play an important role in General Relativity andhave been shown to be an inherent feature of most physically rea-sonable space-times. Despite this, there are many aspects of singu-larities that are not qualitatively or quantitatively understood. Theabstract boundary construction of Scott and Szekeres has proven tobe a flexible tool with which to study the singular points of a man-ifold. The abstract boundary construction provides a ‘boundary’ forany n -dimensional, paracompact, connected, Hausdorff, C ∞ manifold.Singularities may then be defined as entities in this boundary - theabstract boundary. In this paper a topology is defined, for the firsttime, for a manifold together with its abstract boundary. This topol-ogy, referred to as the attached point topology, thereby provides uswith a description of how the abstract boundary is related to the un-derlying manifold. A number of interesting properties of the topologyare considered, and in particular, it is demonstrated that the attachedpoint topology is Hausdorff. Since the inception of the theory of General Relativity, singularities haveplayed an important role. In many instances, they were assumed to be anartefact of an idealised level of symmetry. The powerful singularity theo-rems of Penrose and Hawking [1], however, demonstrated that any genericspace-time with a reasonable distribution of matter satisfying physically rea-sonable conditions would necessarily contain singularities. This implied thatsingularities are therefore an integral part of a space-time.Despite this, without the aid of any additional mathematical structure,we cannot fully answer the question “what is a singularity?”. In part, thisis due to the fact that a singularity is not, technically, part of the manifold,and therefore any description of it purely in terms of the manifold itself willnot be complete. An amount of extra mathematical structure is requiredin order to properly describe a singularity. This extra detail is providedby a boundary construction which gives us a way of rigorously describingthe singular points of a manifold. A boundary construction is therefore anessential tool in properly understanding the global structure of a space-time.Previously, there have been numerous attempts to produce a boundaryconstruction for space-times - most notably the g -boundary of Geroch [2], the b -boundary of Schmidt [3] and the c -boundary of Geroch, Kronheimer andPenrose [4]. All of these boundary constructions, however, suffer from prob-lems and limitations in terms of their application and physical results and, assuch, they do not fully encapsulate all aspects of a singularity. For a detailedsummary of these constructions, see [5], [6] and [7]. The abstract boundary2a-boundary) construction of Scott and Szekeres [8] offers an alternative tothese constructions that is free of many of these issues. It should be notedthat other boundary constructions have been presented recently. Most no-table among these constructions is the iso-causal boundary of Garc´ıa-Parradoand Senovilla [9] which uses an ideology similar to the a-boundary. In ad-dition, the c-boundary continues to be studied and numerous attempts havebeen made to address its known issues. For a summary of these alternativec-boundary constructions, see [10] and [11].When dealing with abstract spaces, there is typically no predefined no-tion of how ‘close’ or ‘separated’ two elements of the space are relative toeach other. A topology provides us with such a notion and is therefore ben-eficial in understanding the structure of these spaces. Although the abstractboundary construction provides us with a collection of abstract boundarypoints, without a topology on it we lack any sense of ‘where’ these pointsare with respect to the manifold in question. Since the abstract boundarypoints represent singularities (among other things), it is of obvious physicalimportance to know where these points are with respect to a space-time, andthus a topology on the manifold together with its abstract boundary is highlydesirable.It should be noted at this point that the b , c and g -boundary construc-tions do have their own topologies. In each case, however, there are problemsassociated with the separation of neighbouring points. The b -boundary, forinstance, has been shown to identify the initial and final singularities of theclosed Friedmann cosmology [12]. It has also been shown that the b -boundaryof a family of space-times, which includes the Friedmann and Schwarzschildsolutions, is non-Hausdorff [13]. Non-Hausdorff g -boundary constructionsalso occur naturally for many space-times. As constructed in [14], these ex-3mple space-times possess boundary points which are not T -separated frommanifold points. The singular points are therefore arbitrarily close to ‘in-terior’ manifold points. The c -boundary likewise suffers from topologicalseparation problems between manifold points and boundary points. Thislack of separation between points appears to be a non-physical property, asit is not clear if non-Hausdorff space-times are realistic [15]. It is there-fore physically desirable for there to exist a natural Hausdorff topology forthe abstract boundary construction. For a more complete discussion of thevarious topological problems associated with each of these three boundaryconstructions, see [6].The main difficulty in constructing a topology for a manifold M and itsabstract boundary B ( M ) is that the abstract boundary points are producedvia embeddings of the manifold. This means that the abstract boundarypoints exist in a space separate to the manifold M . A way of relating theabstract boundary points back to the manifold is therefore required if theyare to be included in open sets that also include elements of M .As usual, there exist a number of possible topologies which can be put on M ∪ B ( M ), some of which will be Hausdorff and first countable. Ideally, wedesire a topology that is physically useful, i.e., the topology should be ableto tell us, for example, ‘where’ in M ∪ B ( M ) the singularities are located.It is therefore essential that the chosen topology somehow relates elementsof the abstract boundary back to M . The topology that is presented insection 4, namely the attached point topology, was developed with this inmind. This topology relies on the idea of an abstract boundary point beingattached to an open set of M , and it represents one of the more naturalpossible constructions. What it means for an abstract boundary point tobe attached, and other related concepts, are discussed in section 3. Various4roperties of the open and closed sets of the attached point topology are thendiscussed in sections 5, 6 and 7.Within this work, we use the following fact frequently and so formallypresent it here for ease of reference. Let g be a Riemannian metric on amanifold M , and let Ω p,q denote the set of piecewise smooth curves in M from p to q . For every curve c ∈ Ω p,q with c : [0 , → M there is a finitepartition 0 = t < t < ... < t k = 1 such that c | [ t i , t i +1 ] is smooth foreach i , 1 ≤ i ≤ k −
1. The Riemannian arc length of c with respect to g isthen defined to be L ( c ) = P k − i =1 R t i +1 t i p g ( c ′ ( t ) , c ′ ( t )) dt , and the Riemanniandistance function, d ( p, q ), between p and q is then defined in terms of this by d ( p, q ) = inf { L ( c ) : c ∈ Ω p,q } ≥
0. The most useful property of this distancefunction is that the open balls defined by B ǫ ( p ) = { q ∈ M : d ( p, q ) < ǫ } form a basis for the manifold topology, and thus the topology induced by theRiemannian metric agrees with the manifold topology [16]. The a-boundary will now be defined. For a more complete discussion of the a-boundary, see [8]. It will be assumed that all manifolds used in the followingwork will be n-dimensional, paracompact, connected, Hausdorff and smooth(i.e., C ∞ ). The manifold topology will be employed throughout the paperunless explicitly stated otherwise. The principle feature of the a-boundaryconstruction is that of an envelopment. Definition 1 (Embedding) The function φ : M → c M is an embedding if φ is a homeomorphism between M and φ ( M ), where φ ( M ) has the subspacetopology inherited from c M . 5 efinition 2 (Envelopment) An enveloped manifold is a triple ( M , c M , φ )where M and c M are differentiable manifolds of the same dimension n and φ isa C ∞ embedding φ : M → c M . The enveloped manifold will also be referredto as an envelopment of M by c M , and c M will be called the envelopingmanifold. Definition 3 (Boundary point) A boundary point p of an envelopment( M , c M , φ ) is a point in the topological boundary of φ ( M ) in c M . The setof all such p is thus given by ∂ ( φ ( M )) = φ ( M ) \ φ ( M ) where φ ( M ) is theclosure of φ ( M ) in c M . The boundary points are then simply the limit pointsof the set φ ( M ) in c M which do not lie in φ ( M ) itself.The characteristic feature of a boundary point is that every open neigh-bourhood of it (in c M ) has non-empty intersection with φ ( M ). Definition 4 (Boundary set) A boundary set B is a non-empty set of suchboundary points for a given envelopment, i.e., a non-empty subset of ∂ ( φ ( M )).It is important to note that different boundary points will arise withdifferent envelopments of M . In order to continue, a notion of equivalencebetween boundary sets of different envelopments is required. This equivalenceis defined in terms of a covering relation. Definition 5 (Covering relation) Given a boundary set B of one envelop-ment ( M , c M , φ ) and a boundary set B ′ of a second envelopment ( M , c M ′ , φ ′ ),then B covers B ′ if for every open neighbourhood U of B in c M there existsan open neighbourhood U ′ of B ′ in c M ′ such that φ ◦ φ ′− ( U ′ ∩ φ ′ ( M )) ⊂ U .
6n essence, this definition says that a sequence of points from within M cannot get close to points of B ′ without at the same time getting close topoints of B . See Fig 1. BU U'B'
PSfrag replacements c M φ ( M ) φ ′ ( M ) c M c M ′ φ ◦ φ ′− ( U ′ ∩ φ ′ ( M )) Figure 1: the boundary set B covers the boundary set B ′ Definition 6 (Equivalent) The boundary sets B and B ′ are equivalent (writ-ten B ∼ B ′ ) if B covers B ′ and B ′ covers B . This definition produces anequivalence relation on the set of all boundary sets. An equivalence class isdenoted by [ B ], where B is a representative of the set of equivalent boundarysets under the covering relation. Definition 7 (Abstract boundary point) An abstract boundary point is thendefined to be an equivalence class [ B ] that has a singleton point p as arepresentative member. Such an equivalence class will then be denoted by[ p ]. The set of all such abstract boundary points of a manifold M will bedenoted by B ( M ) and called the abstract boundary of M . The union of allpoints of a manifold M and its collection of abstract boundary points B ( M )will be labelled as M , i.e., M = M ∪ B ( M ).7 Attached Boundary Points and Sets
In this section, a number of definitions will be presented that describe howthe abstract boundary points of a manifold, M , may be topologically relatedto the points of M . Definition 8 (Attached boundary point) Given an open set U of M and anenvelopment φ : M → c M , then a boundary point p of ∂ ( φ ( M )) is said tobe attached to U if every open neighbourhood N of p in c M has non-emptyintersection with φ ( U ), i.e., N ∩ φ ( U ) = ∅ . See Fig 2. N N' p q
PSfrag replacements c M φ ( M ) φ ( U ) φ ( V )Figure 2: boundary points p and q are attached to the open sets U and V respectively Definition 9 (Attached boundary set) Given an open set U of M and anenvelopment φ : M → c M , then a boundary set B ⊂ ∂ ( φ ( M )) is said tobe attached to U if every open neighbourhood N of B in c M has non-emptyintersection with φ ( U ), i.e., N ∩ φ ( U ) = ∅ . See Fig 3. Note that this does8ot necessarily imply that all points q ∈ B are attached to U , as can be seenin the case illustrated by Fig 3. It does ensure, however, that at least oneboundary point p in B is attached to U . NB PSfrag replacements c M φ ( M ) φ ( U ) φ ( V )Figure 3: boundary set B is attached to the open sets U and V Lemma 10 If B ⊂ ∂ ( φ ( M )) is attached to an open set U of M , then thereexists a p ∈ B such that p is attached to U . Proof:
The boundary set B is attached to U . Therefore, for every openneighbourhood N of B we have that N ∩ φ ( U ) = ∅ . Now assume that no point q ∈ B is attached to U . There therefore exists, for each q , an open neighbour-hood N q of q such that N q ∩ φ ( U ) = ∅ . Now take the union S q ∈ B N q of all ofthe N q . This is an open set containing B such that ( S q ∈ B N q ) ∩ φ ( U ) = ∅ .This contradicts the fact that B is attached to U , and therefore we have thatsome q ∈ B must be attached to U . (cid:3) Because boundary points which are equivalent may appear in a numberof different envelopments, it is necessary to check that definitions (8) and (9)are well defined under the equivalence relation. More specifically, we wish to9how that if a boundary set B ⊂ ∂ ( φ ( M )) is attached to an open set U ⊂ M and there exists a boundary set B ′ ⊂ ∂ ( ψ ( M )) that is equivalent to B , then B ′ is also attached to U . Proposition 11
Let B ⊂ ∂ ( φ ( M )) be attached to an open set U ⊂ M ,and let B ′ ⊂ ∂ ( φ ′ ( M )) be a boundary set of a second envelopment φ ′ . If B ′ covers B , then B ′ is also attached to the open set U ⊂ M . Proof:
Let B ⊂ ∂ ( φ ( M )) be attached to an open set U ⊂ M , and let B ′ ⊂ ∂ ( φ ′ ( M )) be a boundary set which covers B . Assume that B ′ is notattached to U . Thus there exists an open neighbourhood N of B ′ in c M ′ suchthat N ∩ φ ′ ( U ) = ∅ . Since B ′ covers B , for every open neighbourhood N ′ of B ′ there exists an open neighbourhood D of B such that φ ′ ◦ φ − ( D ∩ φ ( M )) ⊂ N ′ . This definition must be true for any neighbourhood N ′ of B ′ , and so wechoose N ′ to be N , so that φ ′ ◦ φ − ( D ∩ φ ( M )) ⊂ N . Since B is attached to U , D ∩ φ ( U ) = ∅ , and since D ∩ φ ( U ) ⊂ D ∩ φ ( M ), φ ′ ◦ φ − ( D ∩ φ ( U )) ⊂ N . Now D ∩ φ ( U ) ⊂ φ ( U ) so that φ ′ ◦ φ − ( D ∩ φ ( U )) ⊂ φ ′ ( U ). Since φ ′ ◦ φ − ( D ∩ φ ( U )) = ∅ , it follows that N ∩ φ ′ ( U ) = ∅ . A contradiction isthus obtained as it was originally assumed that N ∩ φ ′ ( U ) = ∅ . (cid:3) Definition 12 (Attached abstract boundary point) The abstract boundarypoint [ p ] is attached to the open set U of M if the boundary point p isattached to U . Remark:
The abstract boundary point [ p ] is an equivalence class ofboundary sets which are equivalent to p . By proposition (11) the attachedabstract boundary point definition is well defined as any boundary set B such that B ∼ p is also attached to U , i.e., all members of the equivalence10lass [ p ] are attached to U . Proposition 13
Given an open set U of M and an envelopment φ : M → c M , then the set B U of boundary points of ∂ ( φ ( M )) which are attached to U is closed in the induced topology on ∂ ( φ ( M )). Proof: If B U = ∅ or ∂ ( φ ( M )) then, clearly, it is closed in the inducedtopology on ∂ ( φ ( M )). So we will assume that B U = ∅ or ∂ ( φ ( M )). For B U to be closed in the induced topology on ∂ ( φ ( M )), then ∂ ( φ ( M )) \ B U = ∅ must be open in ∂ ( φ ( M )). ∂ ( φ ( M )) \ B U contains the points q ∈ ∂ ( φ ( M ))that are not attached to U , and thus there exists an open neighbourhood N q in c M for each q such that N q ∩ φ ( U ) = ∅ . It follows that N q ∩ B U = ∅ foreach q , because otherwise N q would be a neighbourhood for some p ∈ B U andwould thus intersect φ ( U ). Call the union of all such N q neighbourhoods, A .We therefore have that A ∩ ∂ ( φ ( M )) = ∂ ( φ ( M )) \ B U is an open set in theinduced topology on ∂ ( φ ( M )) and therefore that B U is closed in ∂ ( φ ( M )). (cid:3) Proposition 14
Given an open set U of M and an envelopment φ : M → c M , then the set B U of boundary points of ∂ ( φ ( M )) which are attached to U is closed in c M . See Fig 4. Proof:
Once again, if B U = ∅ or ∂ ( φ ( M )) = c M\ ( c M\ φ ( M ) ∪ φ ( M ))then it is closed in c M , and so we will assume that B U = ∅ or ∂ ( φ ( M )). If B U = B U , then B U is closed in c M . Let x ∈ c M\ B U and assume that x isa limit point of B U . Since φ ( M ) and c M\ φ ( M ) are open sets in c M , it isclear that x φ ( M ) and x c M\ φ ( M ) else otherwise there would exist anopen neighbourhood of x which does not intersect ∂ ( φ ( M )), and thus does11ot intersect B U . It follows that x ∈ ∂ ( φ ( M )). Since x is a limit pointof B U , N x ∩ B U = ∅ for every open neighbourhood N x of x , and therefore N x ∩ φ ( U ) = ∅ for every N x , because every p ∈ B U is attached to U . Thisimplies that x is attached to U which is a contradiction since it was originallyassumed that x ∈ c M\ B U . It therefore follows that B U = B U and thus B U isclosed in c M . (cid:3) PSfrag replacements c M φ ( M ) B U (closed) φ ( U )Figure 4: the closed boundary set B U attached to U A topology on M = M∪B ( M ) may be constructed by defining the open setsin terms of the attached abstract boundary point definition (definition (12)).In keeping with the notion of constructing a natural topology, the open setsof M to which the abstract boundary points are attached are therefore takento be the open sets of the manifold topology.Consider the sets A i = U i ∪ B i , where U i is a non-empty open set of themanifold topology in M and B i is the set of all abstract boundary pointswhich are attached to U i . B i may be the empty set if no abstract boundarypoints are attached to U i . Consider also the sets C i , where each C i is somesubset of the abstract boundary B ( M ). The collection of every C i set is the12et of all subsets of the abstract boundary B ( M ), including all singleton sets { [ p ] } where [ p ] ∈ B ( M ). It will be seen (proposition 21) that the open setsof the topology induced on B ( M ) from the attached point topology on M are precisely the C i sets. Furthermore, it is the presence of certain C i setswhich will ensure that the attached point topology is Hausdorff.Let V be the set comprised of every A i set and every C i set. That is, V = A i = U i ∪ B i C i ⊆ B ( M ) . Lemma 15
Every abstract boundary point [ p ] is attached to an open set U i . Proof:
Let N be any open neighbourhood of p ∈ ∂ ( φ ( M )) in c M . Since p is a boundary point, every open neighbourhood of it has non-empty in-tersection with φ ( M ), and hence N ∩ φ ( M ) is a non-empty open set inthe subspace topology on φ ( M ). In addition, since φ is an embedding, thenon-empty set U i = φ − ( N ∩ φ ( M )) is open in M . Now take any otheropen neighbourhood N ′ of p in c M . Such a neighbourhood will always havenon-empty intersection with N ∩ φ ( M ). This follows from the fact that theintersection of two open sets is another open set: N ′ is an open set that con-tains p , and thus N ∩ N ′ = N ∗ is an open set that also contains p . Because N ∗ is a neighbourhood of p we have that N ∗ ∩ φ ( M ) = ∅ . This then impliesthat ( N ∩ N ′ ) ∩ φ ( M ) = ∅ , i.e., N ′ ∩ φ ( U i ) = ∅ . This then is a statement ofthe attached boundary point condition, i.e., p and thus [ p ] is attached to U i .Every [ p ] is therefore attached to an open set U i in M . (cid:3) Proposition 16
The elements of V form a basis for a topology on M .13 roof: By definition, M is covered by the collection { U i } of open sets in M . Also, by lemma (15), each abstract boundary point is attached to anopen set. The set of open sets in M and their attached abstract boundarypoints, i.e., { A i } , therefore covers M .Now the intersection between two elements of V must be examined. Indoing so, there are three types of intersection that need to be considered.The first is the intersection between A = U ∪ B and A = U ∪ B . Forthis particular intersection, there are several cases to check:1. U ∩ U = ∅ , B ∩ B = ∅ (this includes the cases when B = ∅ or B = ∅ )2. U ∩ U = ∅ , B ∩ B = ∅ U ∩ U = ∅ , B ∩ B = ∅ i) In the first case we have that U ∩ U = ∅ and B ∩ B = ∅ , andtherefore A ∩ A = ( U ∩ U ) ∪ ( B ∩ B ). U ∩ U is another open set U .Assume there exists an abstract boundary point [ p ] that is attached to U .[ p ] is therefore attached to U ([ p ] ∈ B ) and U ([ p ] ∈ B ) which wouldimply that B ∩ B = ∅ . It thus follows that no abstract boundary point isattached to U and so A ∩ A = U ∪ B ∈ V (where B = ∅ ).ii) There are two subcases that need to be considered in the case that U ∩ U = ∅ and B ∩ B = ∅ . The first situation, subcase iia), is depicted inFig 5, and the second situation, subcase iib), is depicted in Fig 6.Subcase iia) refers to the situation where every abstract boundary point[ p ] ∈ B ∩ B is attached to U ∩ U , and subcase iib) refers to the situationwhere B ∩ B = ∅ and there exists a [ p ] ∈ B ∩ B which is not attached to U ∩ U . 14 PSfrag replacements c M φ ( M ) φ ( U ) φ ( U )Figure 5: subcase iia) p PSfrag replacements c M φ ( M ) φ ( U ) φ ( U )Figure 6: subcase iib)Let I = A ∩ A = ( U ∩ U ) ∪ ( B ∩ B ), and Q = ( U ∩ U ) ∪ B ( U ∩ U ) where B ( U ∩ U ) is the set of all abstract boundary points which are attachedto U ∩ U . It may be the case that B ( U ∩ U ) = ∅ (subcase iib)). Otherwise,let [ p ] be an abstract boundary point that is attached to U ∩ U . [ p ] istherefore attached to U ([ p ] ∈ B ) and U ([ p ] ∈ B ). We thus have that[ p ] ∈ B ∩ B and so B ( U ∩ U ) ⊆ B ∩ B , in which case Q ⊆ I . Since Q ∈ V ,any x ∈ Q is contained in an element of V which is a subset of I . Nowsuppose that x ∈ I \ Q , i.e., x is an abstract boundary point [ p ] which is notattached to U ∩ U (subcase iib)). The abstract boundary point [ p ] forms aset C i = { [ p ] } ∈ V , i.e., x ∈ C i ⊆ I . We therefore have that all elements of15 are contained in elements of V , which are subsets of I . iii ) Now consider the final case where U ∩ U = ∅ and B ∩ B = ∅ . SeeFig 7. In this case we have that A ∩ A = B ∩ B , i.e., the intersectionis a collection of abstract boundary points. Since the C i sets are subsets of B ( M ), this collection of abstract boundary points will correspond to a C i set. p N PSfrag replacements c M φ ( M ) φ ( U ) φ ( U )Figure 7: case iii)The next type of intersection to consider is the intersection between A i = U i ∪ B i and C j . If A i ∩ C j = ∅ , then it consists of a collection of abstractboundary points. As before, this collection of abstract boundary points willcoincide with one of the C k sets and we will have that ( U i ∪ B i ) ∩ C j = C k ∈ V .Finally, we consider the intersection between two C j sets. Given any two C j sets, C i and C k , that have non-empty intersection, then C i ∩ C k will be aset of abstract boundary points. Since, by definition, the C j sets are subsetsof B ( M ), there will always exist another set C l such that ( C i ∩ C k ) = C l ∈ V .16his concludes the proof that the elements of V form a basis for a topologyon M . (cid:3) Definition 17 (Attached point topology)
The attached point topology on M is the topology on M which has the basis V .The aim of the attached point topology is to investigate how a givenabstract boundary point is related to the underlying manifold M . This isachieved by hardwiring into the topology, via the definition of an attached ab-stract boundary point (definition 12), what it means for an abstract boundarypoint to be ‘close’ to some part of M . Basically, the location of a particularabstract boundary point [ p ] is fully determined by the set of open sets of M to which it is attached. This provides a natural motivation for our choice oftopology on M with the sets A i = U i ∪ B i , comprised of open sets U i of M together with all abstract boundary points which are attached to U i , formingbasis elements for the attached point topology on M .It should also be noted that the C i sets are an important and necessaryaddition to the basis V . As was seen in case iii) in the proof of proposition 16,where U ∩ U = ∅ and B ∩ B = ∅ (see fig 7), we have that A ∩ A = B ∩ B ,i.e., the intersection is a collection of abstract boundary points. So in anytopology generated from a basis which includes the sets A i , this collection ofabstract boundary points is an open set. This in turn forces the existence ofbasis elements which are collections of abstract boundary points, i.e., the C i sets. 17 Open and Closed Sets in the Attached PointTopology
The open sets of M consist of arbitrary unions of the elements of V . At firstinspection it may seem that an arbitrary open set ( U i ∪ B i ) ∪ ( U j ∪ B j ) ∪ ... is another basis element U k ∪ B k because it is possible to write ( U i ∪ B i ) ∪ ( U j ∪ B j ) ∪ ... as ( U i ∪ U j ∪ ... ) ∪ ( B i ∪ B j ... ) = U k ∪ ( B i ∪ B j ... ). The followingproposition demonstrates, however, that this is not true in general, as theremay be abstract boundary points attached to U k that are not contained in B i ∪ B j ... , i.e., that are not attached to U i , U j , ... . Proposition 18
The sets M and B ( M ) are each both open and closed inthe attached point topology on M . Proof:
For a manifold M , there exists a complete metric d on M suchthat the topology induced by d agrees with the manifold topology of M [16].Choose ǫ >
0, and for each x ∈ M , let U x be the open ball U x = { y ∈ M : d ( x, y ) < ǫ } . Now consider the envelopment φ : M → c M and a boundarypoint p ∈ ∂ ( φ ( M )). We know that p / ∈ φ ( U x ) since d is a complete metricon M . Thus the set c M\ φ ( U x ) is an open neighbourhood of p in c M whichdoes not intersect φ ( U x ), and so p is not attached to U x . It follows that noboundary point p of any envelopment of M is attached to U x , which impliesthat U x has no attached abstract boundary points, i.e., B x = ∅ .Now [ x ∈M A x = [ x ∈M ( U x ∪ B x )= ( [ x ∈M U x ) ∪ ( [ x ∈M B x )18 M ∪ ∅ = M . It follows that M is open in M and thus B ( M ) is closed. Since B ( M ) ⊆B ( M ), B ( M ) is a basis element C i and is therefore open in M , which meansthat M is closed. So the sets M and B ( M ) are each both open and closedin the attached point topology on M . (cid:3) This proposition has demonstrated that M = S x ∈M ( U x ∪ B x ). In gen-eral, M 6 = M ∪ B M , where B M is the collection of all abstract boundarypoints attached to M (i.e., B M = B ( M ) as every boundary point of everyenvelopment of M is attached to M ). It has therefore been demonstratedthat an arbitrary union of basis elements of the topology is, in general, notanother basis element. Example 19
This example illustrates that M is not the only example of anopen set in M that has no attached abstract boundary points and may alsobe written as a union of U i ∪ B i basis sets.Consider M = { ( x, y ) ∈ R : y < } , c M = R and let φ : M → c M be the inclusion map. Let p be the boundary point ( x , p ∈ ∂ ( φ ( M )) isan abstract boundary point representative. Define a sequence { x n } of M by x n ≡ ( x , − n ) so that d ( x n , p ) = 1 /n , where d is the distance functionon c M (which produces the manifold topology of R ). Around every point x n consider the open ball defined by U n = { y ∈ c M : d ( x n , y ) < / ( n + 1) } (see Fig 8). By construction, for each n , U n ⊂ M and thus each U n hasno attached abstract boundary points, i.e., B n = ∅ and U n = U n ∪ B n .Because the sequence { x n } converges to the point p , it follows that every openneighbourhood of p will contain some point x n and therefore will intersectthe open ball U n . The abstract boundary point [ p ] is therefore attached to19 = S n U n , but O may be expressed as a union of non-empty open sets U n in M , each of which does not have any attached abstract boundary points,i.e., O = S n U n = S n ( U n ∪ B n ).PSfrag replacements p = ( x , c M φ ( M ) x U Figure 8: the first 11 elements of the sequence { x n } and their open ballneighbourhoods U n Lemma 20
The singleton abstract boundary point sets, { [ p ] } , are both openand closed in the attached point topology on M . Proof:
For each abstract boundary point [ p ], { [ p ] } ⊆ B ( M ). Thus { [ p ] } = C i , a basis element of V , and is therefore open in the attached point topologyon M . 20ow B ( M ) \{ [ p ] } ⊆ B ( M ) and is therefore a basis element C j of V . ByProposition (18), M is open in the attached point topology on M . The set M ∪ C j = M ∪ ( B ( M ) \{ [ p ] } ) = M\{ [ p ] } is open in M as it is the union oftwo open sets. It follows that { [ p ] } is closed in M .Thus the singleton abstract boundary point sets, { [ p ] } , are both open andclosed in the attached point topology on M . (cid:3) Proposition 21
The open sets of the induced topology on B ( M ) ⊂ M ,where M has the attached point topology, are the C i sets defined in thebasis V . Proof:
Let T M be the attached point topology on M . The subspacetopology on B ( M ) is the collection of sets T B ( M ) = { U ∩ B ( M ) : U ∈ T M } . T M is the collection of arbitrary unions and finite intersections of U j ∪ B j and C i sets. The intersection of these sets U with B ( M ) is therefore thecollection of C i sets. (cid:3) M to M We now consider the inclusion map i : M → M = M ∪ B ( M ) | i ( p ) = p . Itcan be shown that the inclusion map is an embedding. Proposition 22 If M has the attached point topology, then the inclusionmapping i : M → M | i ( p ) = p is an embedding. Proof:
The inclusion mapping i is an embedding if it is a homeomorphismof M onto i ( M ) in the subspace topology on i ( M ) ∩ M . Clearly i is a bijec-tion of M onto i ( M ). Now let T M be the usual topology on M consisting21f the collection of open sets { U i } , T M the attached point topology on M asdefined in section 4 from the basis elements of V , i.e., T M is the collectionof arbitrary unions and finite intersections of the U i ∪ B i and C j sets, and T i ( M ) the subspace topology on i ( M ) ∩ M . The subspace topology T i ( M ) istherefore the collection of sets T i ( M ) = { U k } . Clearly both i and i − are con-tinuous with respect to T M and T i ( M ) . It has thus been demonstrated that i : M → M | i ( p ) = p is a homeomorphism onto its image in the inducedtopology and thus it is an embedding. (cid:3) Because it has been shown that i : M → M | i ( p ) = p is an embedding,we may view M as simply M with the addition of its abstract boundarypoints. This is a pleasing result as one would expect the nature of M to bepreserved in M .The following properties of i ( M ) are readily obtained. Lemma 23
For the inclusion mapping i : M → M | i ( p ) = p , i ( M ) isboth open and closed in the attached point topology on M , i ( M ) = M and ∂ ( i ( M )) = ∅ . Proof:
Since i ( M ) = M , it follows from proposition (18) that i ( M ) isboth open and closed in the attached point topology on M .Because i ( M ) is closed, i ( M ) = i ( M ) = M 6 = M = M ∪ B ( M ). Now ∂ ( i ( M )) = i ( M ) \ i ( M ) = i ( M ) \ i ( M ) = ∅ . (cid:3) In particular, lemma 23 demonstrates that under the inclusion mapping i , M is open and thus B ( M ) is closed in the attached point topology on M as one would desire. 22 Properties of the Attached Point Topology
A number of important properties of the attached point topology will nowbe considered.
Proposition 24
The topological space ( M , T M ), where T M is the attachedpoint topology on M , is Hausdorff. Proof:
Consider two distinct points in M , x and y . Because M is Haus-dorff, there exist open neighbourhoods N x and N y of x and y , respectively,such that N x ∩ N y = ∅ . We now consider whether or not the topological space( M , T M ) is Hausdorff, for while the manifold M is defined to be Hausdorff,( M , T M ) is not necessarily Hausdorff.Given the existence of a complete metric d on M , it was demonstratedin the proof of proposition (18) that, for any v >
0, the open ball { p ∈ M : d ( x, p ) < v } , based at the point x has no attached abstract boundary points.Since N x is an open neighbourhood of x in M , it is possible to choose an ǫ > U x = { p ∈ M : d ( x, p ) < ǫ } , U x ⊂ N x . Nowthe basis element of V , U x ∪ B x , is simply U x since B x = ∅ .Likewise, we can choose an η > U y = { p ∈M : d ( y, p ) < η } , U y ⊂ N y . The basis element U y ∪ B y is simply U y since B y = ∅ .Thus, x ∈ U x ∪ B x , y ∈ U y ∪ B y and ( U x ∪ B x ) ∩ ( U y ∪ B y ) = ( U x ∩ U y ) ⊆ N x ∩ N y = ∅ . The open sets U x ∪ B x and U y ∪ B y are therefore disjoint openneighbourhoods of x and y respectively.Now consider a point x ∈ M and an abstract boundary point [ p ] ∈ B ( M ).As before, for ǫ >
0, the open ball U x = { p ∈ M : d ( x, p ) < ǫ } based at thepoint x has no attached abstract boundary points. Thus, the basis element23f V , U x ∪ B x is simply U x . Now C i = { [ p ] } is also a basis element of V , and( U x ∪ B x ) ∩ C i = U x ∩ C i = ∅ . The open sets U x ∪ B x and C i are thereforedisjoint open neighbourhoods of x and [ p ] respectively.Finally, consider two distinct abstract boundary points [ p ] and [ q ], i.e., p is not equivalent to q . The basis elements of V , C i = { [ p ] } and C j = { [ q ] } ,are disjoint open neighbourhoods of [ p ] and [ q ] respectively, since [ p ] and [ q ]are different equivalence classes.Having considered all possible combinations of different types of elementsof M , namely x, y ∈ M , x ∈ M and [ p ] ∈ B ( M ), and [ p ] , [ q ] ∈ B ( M ),we have thereby demonstrated that the topological space ( M , T M ) is indeedHausdorff. (cid:3) We shall also check if the attached point topology is first countable.
Proposition 25
The attached point topology on M is first countable. Proof:
A topological space X is said to be first countable if for each x ∈ X , there exists a sequence U , U ,... of open neighbourhoods of x suchthat for any open neighbourhood, V , of x , there exists an integer, i , suchthat U i ⊆ V .For X = M , we firstly consider the case where x ∈ M . Given theexistence of a complete metric d on M , we know from the proof of proposition(18) that, for n ∈ N , the open balls U n = { p ∈ M : d ( x, p ) < /n } basedat the point x have no attached abstract boundary points. The sets U n ∪ B n = U n are basis elements of V , and so U , U ,... is a sequence of openneighbourhoods of x .Let V be an open neighbourhood of x in M . V is an arbitrary union24r finite intersection of basis elements A i and C j and therefore has the form V = U ∪ B where U is an open set in M , x ∈ U , and B ⊆ B ( M ) (wherepossibly B = ∅ ). It is possible to choose an n ∈ N , such that, for the openball U n , U n ⊂ U . Thus U n ⊆ V . We have therefore shown that M is firstcountable at x , for all x ∈ M .Now we consider an abstract boundary point [ p ] ∈ B ( M ). For each n ∈ N , define C n = { [ p ] } . The basis elements C n form a sequence, C , C ,...of open neighbourhoods of [ p ]. Now if V is an open neighbourhood of [ p ] in M , then [ p ] ∈ V and C n = { [ p ] } ⊆ V . This means that M is first countableat [ p ], for all [ p ] ∈ B ( M ).We have thereby shown that the attached point topology for M is firstcountable. (cid:3) The abstract boundary construction is a mathematical tool used to find andclassify the boundary features of a space-time, including any singularities.The ability to classify singular points, however, represents only half of thepicture. In order to fully understand the significance of a particular singular-ity, we must also understand how that singularity is connected to the originalspace-time. The attached point topology, defined on the union of a manifoldwith its abstract boundary, provides us with one such description, and hasthe advantage that its construction flows naturally from the definitions ofthe abstract boundary construction itself.It was shown that the attached point topology is Hausdorff which is con-sidered an important ingredient for a workable boundary definition. One ofthe key elements in the attached point topology being Hausdorff is that every25bstract boundary point is an open set. As a consequence of this, every ab-stract boundary point may be separated from every other abstract boundarypoint as well as every point of the manifold M . Therefore, as well as ensur-ing that V is a basis, the C i sets also serve to guarantee that the attachedpoint topology is Hausdorff. The intention of the attached point topologywas to construct a Hausdorff topology which flows naturally from the at-tached abstract boundary point definition (definition 12). The defined C i sets represent a simple solution to the problem of defining a collection of setsof abstract boundary points which ensure that V is a basis for a Hausdorfftopology.The ‘location’ of an abstract boundary point, e.g., a singularity, is hard-wired into the attached point topology through the basis elements A i = U i ∪ B i . Every abstract boundary point is attached to a non-empty openset U i ⊂ M (Lemma (15)). This means that, for a given abstract boundarypoint, for a boundary point representative p occurring in an envelopment φ ,the open set image φ ( U i ) of U i under φ extends all the way out to p in thisenvelopment. Thus the boundary point p is ‘close’ to the open set U i . Sincethis must also be true for every boundary point representative of the abstractboundary point, we thereby have an a priori knowledge of which particularopen sets of M are ‘close’ to our given abstract boundary point. This givesus the location for the boundary features such as singularities.The fact that the attached point topology is naturally Hausdorff is apleasing result as, unlike a number of the other boundary constructions, wedo not have to be concerned with specific space-time examples where we loseseparability, as was discussed in the introduction. In addition, we do not needto consider further conditions on the manifold itself or its boundary in orderto ensure that the topology on M is Hausdorff. In the case of the c -boundary,26or instance, it has been suggested that extra causality conditions on themanifold, such as it being stably causal, would ensure that the resultingtopology on the boundary is Hausdorff [18], [19].In a forthcoming paper, a second topology will be considered for M = M ∪ B ( M ) in which the abstract boundary B ( M ) is a closed set. As a con-sequence of this, however, a number of the abstract boundary points becomeinseparable, and thus the Hausdorff property is lost in general. Separabilityis lost in a very particular way, however, to the extent that this lack of separa-bility may contain additional information about the abstract boundary itself. Acknowledgements
The authors would like to thank Benjamin Whale for useful discussionsrelating to this paper.
References [1] Hawking S W and Penrose R 1970 The Singularities of GravitationalCollapse and Cosmology
Proc. R. Soc. Lond. A
J. Math. Phys. Gen. Rel. and Grav. Proc. R. Soc. Lond. A.
The Large Scale Structure of Space-Time (New York: Cambridge University Press)[6] Ashley M J 2002 Singularity Theorems and the Abstract Bound-ary Construction
PhD Thesis
The Australian National University(http://thesis.anu.edu.au/public/adt-ANU20050209.165310/index.html)[7] Whale B E 2010 Foundations of and Applicationsfor the Abstract Boundary Construction for Space-Time
PhD Thesis
The Australian National University(http://dspace.anu.edu.au/bitstream/1885/49393/1/01front.pdf)[8] Scott S M and Szekeres P 1994 The Abstract Boundary - A New Ap-proach to Singularities of Manifolds
J. Geom. Phys. Class.Quantum Grav. Class. Quantum Grav. R1–R84[11] Flores J L 2007 The Causal Boundary of Spacetimes Revisited
Commun.Math. Phys. b -boundary of the Closed Friedmann-Model Commun. Math. Phys. J. Math.Phys. J. Math. Phys. Commun. Math.Phys. Global Lorentzian Geometry (New York: Marcel Dekker, Inc) p3[17] Munkres J 2000
Topology (New Jersey: Prentice Hall, Inc)[18] Szabados L 1988 Causal Boundary for Strongly Causal Space-times