The Strongly 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 2014 Class. QuantumGrav. 31(12) 125004 doi: 10.1088/0264-9381/31/12/125004
The Strongly Attached Point Topology of the Abstract BoundaryFor Space-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
The abstract boundary construction of Scott and Szekeres pro-vides a ‘boundary’ for any n -dimensional, paracompact, connected,Hausdorff, C ∞ manifold. Singularities may then be defined as objectswithin this boundary. In a previous paper [1], a topology referred toas the attached point topology was defined for a manifold and its ab-stract boundary, thereby providing us with a description of how theabstract boundary is related to the underlying manifold. In this paper,a second topology, referred to as the strongly attached point topology,is presented for the abstract boundary construction. Whereas the ab-stract boundary was effectively disconnected from the manifold in theattached point topology, it is very much connected in the stronglyattached point topology. A number of other interesting properties ofthe strongly attached point topology are considered, each of whichsupport the idea that it is a very natural and appropriate topology fora manifold and its abstract boundary. In the paper ‘The attached point topology of the abstract boundary for space-time’ [1], a topology for a manifold M and its collection of abstract boundarypoints B ( M ) was constructed. The topology, referred to as the attachedpoint topology, represents one of the more natural topologies that can beplaced upon the abstract boundary. It was produced via obvious extensionsto the abstract boundary point definitions, and did not require any additionalconditions to be placed upon the manifold or its boundary. However, it wasdemonstrated that it was possible to separate the manifold and its abstractboundary by disjoint open sets of the attached point topology. The manifoldand its abstract boundary were therefore disconnected from one another insome sense. Even so, the fact that the attached point topology was Hausdorffwas a pleasing result and suggested that the attached point topology was agood starting point in producing a topology that is more descriptive of thetopological relationship between a manifold and its abstract boundary.Because the abstract boundary is produced via embeddings of the mani-fold, the abstract boundary exists in a space separate to that of the manifold.A topology on M ∪ B ( M ) should therefore connect the abstract boundary tothe underlying manifold M . As noted previously, the attached point topol-ogy provided one such description of the topological relationship between M and B ( M ). This topology relied on the notion of abstract boundarypoints being ‘close’, in some sense, to open sets of the manifold M . Ab-2tract boundary points that were close to an open set of M were said to be‘attached’ to that open set, and thus the ‘location’ of an abstract boundarypoint could be described relative to the known topology of M . Because it ispossible to separate the manifold and its abstract boundary from one anotherby disjoint open sets of the attached point topology, it can be said that theattached point topology does not fully integrate B ( M ) with the structure of M . The fact that M and B ( M ) can be separated in this way is a resultof there being too many open sets in the attached point topology. In thispaper, a new topology referred to as the strongly attached point topologyis considered. The strongly attached point topology is defined similarly tothe attached point topology but has one additional restriction. This restric-tion limits the ‘type’ of open set in M to which an abstract boundary pointmay be considered to be ‘close to’, and hence there are comparatively feweropen sets in the strongly attached point topology. The main consequence ofthis is that every open neighbourhood of an abstract boundary point neces-sarily contains some part of the manifold M , i.e., the abstract boundary istopologically inseparable from the underlying manifold M .In section 2, the abstract boundary will again be defined as a matterof convenience. The strongly attached abstract boundary point definitionis developed in 3, which describes how an abstract boundary point may berelated back to M . The strongly attached point topology, which utilises thepreviously mentioned definition is presented in section 4. Various propertiesof the topology are then discussed in sections 5, 6, 7 and 8.We refer the reader interested in the g-boundary, b-boundary and c-boundary to [2], [3] and [4]. For those interested in the more recent causalboundary, see [5], [6], [7], [8], [9] and [10].Within this work, we use the following fact frequently and so formally3resent 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 [11]. For the convenience of the reader, we will provide the definition of the a-boundary in this section. For a more complete discussion of the a-boundary,see [12], [13], [14] and [15]. It will be assumed that all manifolds used in thefollowing work will be n-dimensional, paracompact, connected, Hausdorffand smooth (i.e., C ∞ ). The manifold topology will be employed throughoutthe paper unless explicitly stated otherwise. The principle feature of thea-boundary construction 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 . Definition 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 is4 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 set ofall such points p is thus given by ∂ ( φ ( M )) = φ ( M ) \ φ ( M ) where φ ( M ) isthe closure of φ ( M ) in c M . The boundary points are then simply the limitpoints of 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 ′ , denoted B ⊲ B ′ , if for every open neighbourhood U of B in c M there exists an open neighbourhood U ′ of B ′ in c M ′ such that φ ◦ φ ′− ( U ′ ∩ φ ′ ( M )) ⊂ U . In 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 to5oints of B . See figure 1. BU U'B'
PSfrag replacements c M φ ( M ) φ ′ ( M ) c M d 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 ⊲ B ′ and B ′ ⊲ B . This definition produces an equivalencerelation on the set of all boundary sets. An equivalence class is denoted by[ B ], where B is a representative of the set of equivalent boundary sets underthe covering relation. Definition 7 (Abstract boundary point and abstract boundary) An abstractboundary point is defined to be an equivalence class [ B ] that has a singletonboundary point { p } as a representative member. Such an equivalence classwill then simply be denoted by [ p ]. The set of all such abstract boundarypoints of a manifold M will be denoted by B ( M ) and called the abstractboundary of M . The union of all points of a manifold M and its collectionof abstract boundary points B ( M ) may then be labelled as M , i.e., M = M ∪ B ( M ). Definition 8 (Covered abstract boundary point) An abstract boundarypoint [ p ] covers an abstract boundary point [ q ], denoted [ p ] ⊲ [ q ], if the rep-resentative singleton boundary point { p } covers the representative singleton6oundary point { q } . The attached point topology was defined by topologically relating the ab-stract boundary points of a manifold M back to the points of M via thedefinition of an attached boundary point. We include this definition and thedefinition of an attached boundary set for the benefit of the reader. Definition 9 (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 ) = ∅ . Definition 10 (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 to be attached to U if every open neighbourhood N of B in c M has non-emptyintersection with φ ( U ), i.e., N ∩ φ ( U ) = ∅ .The strongly attached point topology also relies on the notion of an ab-stract boundary point being attached to an open set of M , but the mannerin which the abstract boundary point is attached is different. Definition 11 (Strongly attached boundary point) Given an open set U of M and an envelopment φ : M → c M , then a boundary point p of ∂ ( φ ( M ))is said to be strongly attached to U if there exists an open neighbourhood N of p in c M such that N ∩ φ ( M ) ⊆ φ ( U ). See figure 2.7 PSfrag replacements c M φ ( M ) N ∩ φ ( M ) φ ( U ) N Figure 2: a boundary point p strongly attached to the open set U Lemma 12
If the boundary point p is strongly attached to the open set U then p is attached to U . Proof:
Consider an envelopment φ : M → c M and a boundary point p ∈ ∂ ( φ ( M )). Suppose that p is strongly attached to the open set U ⊂M . There therefore exists an open neighbourhood N of p in c M such that N ∩ φ ( M ) ⊆ φ ( U ). Any other open neighbourhood of p will have non-emptyintersection with N ∩ φ ( M ). This follows from the fact that the intersectionof two open sets is another open set: N ′ is an open set that contains p , andthus N ∩ N ′ = N ∗ is an open set that also contains p . Because N ∗ is aneighbourhood of the boundary point p we have that N ∗ ∩ φ ( M ) = ∅ . Thisimplies that ( N ∩ N ′ ) ∩ φ ( M ) = ∅ , and so N ′ ∩ φ ( U ) = ∅ . This is a statementof the attached boundary point condition, i.e., p is attached to U . (cid:3) The requirement that there exists an open neighbourhood N of p in c M such that N ∩ φ ( M ) ⊆ φ ( U ) removes the possibility of boundary points beingstrongly attached to open sets like those depicted in figure 3, i.e., open sets8hat are shaped more like a wedge and which have minimal ‘contact’ withthe boundary of the particular envelopment. If a boundary point is stronglyattached to an open set U then that set U will always be more ‘spread out’along the boundary under the given envelopment. p PSfrag replacements c M φ ( M ) φ ( U ) N Figure 3: a boundary point p attached, but not strongly attached, to an openset U of M Lemma 13
If a boundary point p ∈ ∂ ( φ ( M )) is strongly attached to theopen sets U and U , then U ∩ U = ∅ . Furthermore, p is strongly attachedto U ∩ U . See figure 4. Proof:
The boundary point p is strongly attached to U and so thereexists an open neighbourhood N of p in c M such that N ∩ φ ( M ) ⊆ φ ( U ).It is also strongly attached to U and so there exists an open neighbourhood N ′ of p in c M such that N ′ ∩ φ ( M ) ⊆ φ ( U ). Now N and N ′ are both openneighbourhoods of p and so their intersection is another open neighbourhoodof p . In addition, p is a boundary point, and so every open neighbourhoodof p has non-empty intersection with φ ( M ). We therefore have ( N ∩ N ′ ) ∩ φ ( M ) = ∅ . Now, since N ∩ φ ( M ) ⊆ φ ( U ) and N ′ ∩ φ ( M ) ⊆ φ ( U ), we9ave that ( N ∩ N ′ ) ∩ φ ( M ) ⊆ φ ( U ) and ( N ∩ N ′ ) ∩ φ ( M ) ⊆ φ ( U ). Thisimplies that φ ( U ) ∩ φ ( U ) = ∅ and therefore that U ∩ U = ∅ . Moreover,( N ∩ N ′ ) ∩ φ ( M ) ⊆ φ ( U ) ∩ φ ( U ) = φ ( U ∩ U ) from which it follows that p is strongly attached to U ∩ U . (cid:3) p PSfrag replacements c M φ ( M ) N ∩ N ′ φ ( U ) φ ( U )Figure 4: the boundary point p is strongly attached to U ∩ U Definition 14 (Strongly attached boundary set) Given an open set U of M and an envelopment φ : M → c M , then a boundary set B ⊂ ∂ ( φ ( M )) is saidto be strongly attached to U if there exists an open neighbourhood N of B in c M such that N ∩ φ ( M ) ⊆ φ ( U ). See figure 5. Lemma 15 If B ⊂ ∂ ( φ ( M )) is strongly attached to the open set U ⊂ M then B is attached to U . Proof:
The proof of this is identical to the proof of lemma 12, exceptthat we are dealing with open neighbourhoods of a boundary set rather thanopen neighbourhoods of a boundary point. (cid:3) PSfrag replacements c M φ ( M ) Nφ ( U ) N ∩ φ ( M )Figure 5: a boundary set B is strongly attached to an open set U of M Lemma 16
A boundary set B ⊂ ∂ ( φ ( M )) is strongly attached to an openset U ⊆ M if and only if every boundary point p ∈ B is strongly attachedto U . Proof: ( ⇒ ) Let φ : M → c M be an envelopment, and B ⊂ ∂ ( φ ( M )) aboundary set that is strongly attached to the open set U ⊆ M . Because B is strongly attached to U , there exists an open neighbourhood, N , of B suchthat N ∩ φ ( M ) ⊆ φ ( U ). N is an open neighbourhood of every boundarypoint p ∈ B . Clearly then, every p ∈ B is strongly attached to U .( ⇐ ) If every boundary point p ∈ B is strongly attached to U , then thereexists an open neighbourhood, N p , of each p such that N p ∩ φ ( M ) ⊆ φ ( U ).The union N B of every N p , i.e., N B = S p ∈ B N p , is an open neighbourhood of B such that N B ∩ φ ( M ) ⊆ φ ( U ). The boundary set B is therefore stronglyattached to U . (cid:3) Lemma 17
If a boundary set B is strongly attached to the open sets U and U , then U ∩ U = ∅ . Furthermore, B is strongly attached to U ∩ U .11 roof: The proof follows from lemma 13, except that we are dealing withopen neighbourhoods of a boundary set, rather than open neighbourhoodsof a boundary point. (cid:3)
Because boundary points which are equivalent may appear in a numberof different envelopments, it is necessary to check that definitions 11 and 14are well defined under the equivalence relation. More specifically, we wish toshow that if a boundary set B ⊂ ∂ ( φ ( M )) is strongly attached to an openset U ⊂ M and there exists a boundary set B ′ ⊂ ∂ ( ψ ( M )) that is equivalentto B , then B ′ is also strongly attached to U . Proposition 18
Let B ⊂ ∂ ( φ ( M )) be strongly attached to an open set U ⊂ M , and let B ′ be a boundary set of a second envelopment φ ′ : M → c M ′ .If B ⊲ B ′ , then B ′ is also strongly attached to U . Proof:
The boundary set B ⊂ ∂ ( φ ( M )) is strongly attached to the openset U . There therefore exists an open neighbourhood N of B such that N ∩ φ ( M ) ⊆ φ ( U ). Now, since B ⊲B ′ , we have that φ ◦ φ ′− ( N ′ ∩ φ ′ ( M )) ⊂ N ,where N ′ is an open neighbourhood of B ′ in c M ′ . It follows that φ ◦ φ ′− ( N ′ ∩ φ ′ ( M )) ⊂ N ∩ φ ( M ) ⊆ φ ( U ), and therefore N ′ ∩ φ ′ ( M ) ⊆ φ ′ ( U ), i.e., B ′ isstrongly attached to U . (cid:3) Definition 19 (Strongly attached abstract boundary point) The abstractboundary point [ p ] is said to be strongly attached to the open set U of M ifthe boundary point p is strongly attached to U .The abstract boundary point [ p ] is an equivalence class of boundary setswhich are equivalent to { p } . By proposition 18 the strongly attached abstract12oundary point definition is well defined as any boundary set B such that B ∼ p is also strongly attached to U , i.e., all members of the equivalenceclass [ p ] are strongly attached to U .Also, by lemma 15 it is clear that if an abstract boundary point [ p ] isstrongly attached to an open set U of M , then [ p ] is also attached to U . Proposition 20
Consider an open set U of M and an envelopment φ : M → c M . Let B U be the set of boundary points of ∂ ( φ ( M )) which arestrongly attached to U . The set B U is closed in c M if and only if the limitpoints of B U are strongly attached to U . Proof:
Since B U ⊂ ∂ ( φ ( M )), any limit point of B U in c M will also lie in ∂ ( φ ( M )). By definition, B U is closed in c M if and only if B U contains all itslimit points. Clearly, B U contains all its limit points if and only if the limitpoints of B U are strongly attached to U , from which the result follows. (cid:3) Corollary 21
Consider an open set U of M and an envelopment φ : M → c M . Let B U be the set of boundary points of ∂ ( φ ( M )) which are stronglyattached to the U . The set B U is closed in the induced topology on ∂ ( φ ( M ))if and only if the limit points of B U are strongly attached to U . Proof:
Since ∂ ( φ ( M )) is closed in c M and B U ⊂ ∂ ( φ ( M )), the set B U isclosed in the induced topology on ∂ ( φ ( M )) if and only if B U is closed in c M .The result then follows directly from proposition 20. (cid:3) In general, B U will not be closed in ∂ ( φ ( M )) or c M because not all thelimit points of B U are necessarily strongly attached to U . See figure 6 andfigure 7. As was shown in proposition 13 and proposition 14 of [1], however,13he limit points of B U are always attached to U .PSfrag replacements c M φ ( M ) B U (not closed) φ ( U )Figure 6: the boundary set B U is not closed because two limit points of B U (one at each end) are not strongly attached to U PSfrag replacements c M φ ( M ) B U (closed) φ ( U )Figure 7: the boundary set B U is closed because all limit points of B U arestrongly attached to U Similarly to the attached point topology, a basis for a topology on M = M ∪ B ( M ) may be constructed by defining the open sets in terms of the14trongly attached abstract boundary point definition (definition 19). Again,in keeping with the notion of constructing a natural topology, the open sets of M to which the abstract boundary points are strongly attached are thereforetaken to 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 strongly attached to U i . B i may be the empty set if no abstractboundary points are strongly attached to U i . Let W be the set comprised ofevery A i set. That is, W = { A i = U i ∪ B i } . Given that the strongly attached boundary point definition limits the‘type’ of open set U i in M to which a boundary point may be stronglyattached, it is important to know whether or not every abstract boundarypoint is strongly attached to an open set U i in M . In other words, if aboundary point p were attached to a ‘wedge’ shaped open set U in M likethat depicted in figure 3, does there always exist another open set U ′ in M to which p is strongly attached? A brief analysis reveals that the answer tothis question is yes - every boundary point p is strongly attached to an openset U in M . Lemma 22
Every abstract boundary point [ p ] is strongly attached to anopen set U in M . Proof:
For the envelopment φ : M → c M , p ∈ ∂ ( φ ( M )), let N be anyopen neighbourhood of p in c M . Since p is a boundary point, N ∩ φ ( M ) isnon-empty. In addition, since φ is an embedding, the non-empty set U = φ − ( N ∩ φ ( M )) is an open set in M . We then have that p is strongly attached15o U = φ − ( N ∩ φ ( M )) because there exists an open neighbourhood N of p in c M such that N ∩ φ ( M ) ⊆ N ∩ φ ( M ). (cid:3) Proposition 23
The elements of W form a basis for a topology on M . Proof:
By definition, M is covered by the collection { U i } of open sets in M . Also, by lemma 22 each abstract boundary point is strongly attached toan open set U i in M . The set of open sets in M and their strongly attachedabstract boundary points, i.e., { A i } , therefore covers M .The intersection between two elements of W must be examined. Considerthe intersection between A = U ∪ B and A = U ∪ B . In considering thisintersection, there are three subcases 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 = ∅ . Since B ∩ B = ∅ , A ∩ A = U ∩ U = U which is an open set in M . If theabstract boundary point [ p ] is strongly attached to U , then [ p ] is stronglyattached to U ([ p ] ∈ B ) and [ p ] is strongly attached to U ([ p ] ∈ B ) whichwould imply that B ∩ B = ∅ . It follows that B = ∅ , where B is theset of abstract boundary points that are strongly attached to U , and thus A ∩ A = U ∪ B ∈ W . ii ) For this case, A ∩ A = ( U ∩ U ) ∪ ( B ∩ B ) = U ∪ ( B ∩ B ). If[ p ] ∈ B ∩ B , it is strongly attached to both U and U , so by lemma 13,16 p ] is strongly attached to U ∩ U = U , i.e., [ p ] ∈ B . Thus B ∩ B ⊆ B .Now if [ p ] ∈ B , it is strongly attached to U = U ∩ U and so is stronglyattached to both U and U . That is, [ p ] ∈ B ∩ B and so B ⊆ B ∩ B . Itfollows that A ∩ A = U ∪ B ∈ W . iii ) This case cannot exist by lemma 13. Specifically, if B ∩ B = ∅ , thenthere exist abstract boundary points that are strongly attached to both U and U , and hence U ∩ U = ∅ .The intersection A ∩ A = ( U ∩ U ) ∪ ( B ∩ B ) is therefore alwaysanother element of W . Thus the elements of W form a basis for a topologyon M . (cid:3) Definition 24 (Strongly attached point topology) The strongly attachedpoint topology on M is the topology which has the basis W .The attached point topology required that sets of abstract boundarypoints be added to the collection of basis sets V . This was done to ensurethat the sets of V did in fact define a basis for a topology on M . Becausethere exist basis sets A i = U i ∪ B i and A j = U j ∪ B j of the attached pointtopology such that A i ∩ A j = B i ∩ B j , i.e., U i ∩ U j = ∅ , the C i sets which arecollections of abstract boundary points must also be included in the collec-tion V of basis sets. The collection W of basis sets for the strongly attachedpoint topology, however, does not require the addition of such sets of abstractboundary points. This is a direct consequence of lemma 13. The non-emptyintersection of any two A i sets will necessarily contain points of M , andtherefore it is impossible that a collection of abstract boundary points can17e produced by considering intersections of A i sets. It is for this reasonthat the topology is referred to as the strongly attached point topology -the abstract boundary B ( M ) is firmly affixed to the manifold M and hasbecome, topologically speaking, an integral part of the larger space M . Inother words, any open neighbourhood of an abstract boundary point [ p ] willnecessarily include some part of M . The open sets of M consist of arbitrary unions of the elements of W . As inthe case of the attached point topology, it may again seem that an arbitraryopen set ( U i ∪ B i ) ∪ ( U j ∪ B j ) ∪ ... is another basis element U k ∪ B k . Again,this is not true in general. Example 25
Consider M = { ( x, y ) ∈ R : y < } , c M = R and let φ : M → c M be the inclusion map. Let p ∈ ∂φ ( M ) be the boundary point(0 , p ] is the associated abstract boundary point. Define a sequence { x n } in M by x n ≡ (0 , − n ). Around each x n define an open set U n = { ( x, y ) : − < x < , − < y < − n +1 } . See figure 8. By construction, in c M , forany n , U n ⊂ M and thus U n has no strongly attached abstract boundarypoints, i.e., B n = ∅ and U n = U n ∪ B n = A n . Take an open ball B ( p ) ofradius around p and consider some a ∈ B ( p ) ∩ M . Because { x n } → p , a will be contained in some U n . Since this is true for every a ∈ B ( p ) ∩ M ,it follows that B ( p ) ∩ M ⊂ S n U n . The abstract boundary point [ p ] istherefore strongly attached to the open set S n U n = O but O is the unionof non-empty open sets U n in M , each of which does not have any strongly18ttached abstract boundary points, i.e., O = S n U n = S n ( U n ∪ B n ). Since O ⊂ M and [ p ] / ∈ O , O / ∈ W .PSfrag replacements p = (0 , c M φ ( M ) x U Figure 8: the first 8 elements of the sequence { x n } and their open neighbour-hoods U n Proposition 26
The set M is open, and the set B ( M ) is closed in thestrongly 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 [11].Choose ǫ >
0, and for each x ∈ M , let U x be the open ball U x = { y ∈ M :19 ( 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 metric on M and so φ ( U x ) ⊂ φ ( M ). Thus the set c M\ φ ( U x ) is an open neighbourhoodof p in c M which does not intersect φ ( U x ), and so p is not attached to U x . Bylemma 12, p is also not strongly attached to U x . It follows that no boundarypoint p of any envelopment of M is strongly attached to U x , which impliesthat U x has no strongly 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 )= M ∪ ∅ = M . It follows that M is open in M and thus B ( M ) is closed because M\B ( M ) = M is open. (cid:3) Proposition 27
The set B ( M ) is not open, and the set M is not closed inthe strongly attached point topology on M . Proof:
Consider any open neighbourhood of an abstract boundary point[ p ] ∈ B ( M ) in M . Every open set of M is a union of basis sets. Becauseevery basis set contains a non-empty open subset of M , every open set inthe strongly attached point topology will contain a non-empty open subsetof M . Any open set that contains [ p ] ∈ B ( M ) will therefore necessarilycontain some open subset of M as well, and thus B ( M ) cannot be open.Since B ( M ) = M\M is not open, M is not closed. (cid:3) Proposition 28
The manifold topology on M and the topology induced on20 by the strongly attached point topology on M are the same. Proof:
Let U = ∅ be an open set in M in the manifold topology. Theset A = U ∪ B , where B is the set of all abstract boundary points which arestrongly attached to U , is an open set in the strongly attached point topologyof M . Now U = A ∩ M and thus U is an open set in the topology induced on M by the strongly attached point topology on M . Now let U = ∅ be an openset in the topology induced on M by the strongly attached point topologyon M . So U = V ∩ M where V is an open set in the strongly attached pointtopology on M . The set V can be expressed as a union of elements of W , i.e., V = S i ∈ I ( U i ∪ B i ). Thus U = ( S i ∈ I ( U i ∪ B i )) ∩M = ( S i ∈ I U i ) ∩M = S i ∈ I U i ,and so U is a union of non-empty open sets of the manifold topology on M and is, therefore, itself an open set of the manifold topology on M . (cid:3) Corollary 29 If V is an open neighbourhood of the abstract boundary point[ p ] in M , then V ∩ M 6 = ∅ . Proof:
This result follows immediately from the proof of proposition 27. (cid:3)
Proposition 27 and corollary 29 summarise the key difference betweenthe attached point topology and the strongly attached point topology. Incontrast to the attached point topology, it is not possible to construct anopen set of abstract boundary points in the strongly attached point topologythat does not also intersect M . This is demonstrated by corollary 29. It wasalso previously shown that the set of abstract boundary points is both openand closed with respect to the attached point topology, thus resulting in theset of all abstract boundary points being disconnected from M . Proposition217 shows that this is not the case for the strongly attached point topology.As discussed in the introduction, the abstract boundary points can thereforebe regarded as being firmly affixed to the manifold M with respect to thestrongly attached point topology. Corollary 30 If V is an open neighbourhood of the abstract boundary point[ p ] in M , then [ p ] is strongly attached to the open set U = ∅ of M , where U ≡ V ∩ M . Proof: V is an open neighbourhood of [ p ] in M and thus it may be writ-ten as a union of basis sets U i ∪ B i , where [ p ] is an element of at least one ofthe B i sets. It follows that U = V ∩ M = S i U i . By proposition 28, we knowthat U is an open set of M , and by corollary 29, we have that S i U i = ∅ .Moreover, [ p ] is strongly attached to one of the U i sets and thus [ p ] is stronglyattached to U where U = ∅ . (cid:3) We will next consider if the singleton abstract boundary point sets { [ p ] } are open or closed in the strongly attached point topology. Before addressingthat question, however, three useful results are established. Proposition 31
For abstract boundary points [ p ] and [ q ], [ p ] ⊲ [ q ] if and onlyif [ q ] is strongly attached to every open set, U ⊆ M , to which [ p ] is stronglyattached. Proof: ( ⇐ ) Suppose that [ q ] is strongly attached to every open set, U ⊆ M , to which [ p ] is strongly attached. Consider the boundary point p of the envelopment φ : M → c M , and the boundary point q of the en-velopment φ ′ : M → c M ′ . Let N be an open neighbourhood of p in c M .22t follows that [ p ] is strongly attached to φ − ( N ∩ φ ( M )). Because [ q ] isstrongly attached to every open set U ⊆ M to which [ p ] is strongly at-tached, [ q ] is strongly attached to φ − ( N ∩ φ ( M )), for all N . For every N there therefore exists an open neighbourhood W of q in c M ′ such that W ∩ φ ′ ( M ) ⊆ φ ′ ◦ φ − ( N ∩ φ ( M )), i.e., { p } covers { q } , and thus [ p ] ⊲ [ q ].( ⇒ ) This follows immediately from proposition 18. (cid:3) Corollary 32
For abstract boundary points [ p ] and [ q ], [ p ] ⊲ [ q ] if and onlyif every open neighbourhood of [ p ] in M also contains [ q ], where M has thestrongly attached point topology. Proof: ( ⇐ ) Suppose that every open neighbourhood of [ p ] in M also con-tains [ q ], where M has the strongly attached point topology. Also supposethat [ p ] is strongly attached to the open set U ⊆ M . Now define the set V = U ∪ B U , where B U is the set of all abstract boundary points which arestrongly attached to U . It is clear that V is an open set in M in the stronglyattached point topology, [ p ] ∈ V , and thus V is an open neighbourhood of[ p ] in M . By assumption, [ q ] ∈ V , and therefore [ q ] is strongly attached tothe open set U ⊆ M . It follows from proposition 31 that [ p ] ⊲ [ q ].( ⇒ ) Suppose now that [ p ] ⊲ [ q ]. Let V be an open neighbourhood of[ p ] in M , where M has the strongly attached point topology. The set V can be expressed as a union of elements of W , i.e., V = S i ∈ I ( U i ∪ B i ). Theabstract boundary point [ p ] lies in at least one set B i and is therefore stronglyattached to the open set U i of M . It follows from proposition 31 that [ q ]is also strongly attached to U i , and so [ q ] ∈ B i . Thus V is also an open23eighbourhood of [ q ]. (cid:3) Corollary 33
The closure in M of an abstract boundary point [ p ] is { [ p ] } = { [ p ] } ∪ { [ x ] : [ x ] ∈ B ( M ) , [ x ] ⊲ [ p ] } . Proof:
From proposition 26, B ( M ) is closed in the strongly attachedpoint topology on M . Since [ p ] ∈ B ( M ), it follows that { [ p ] } ⊆ B ( M ). Nowconsider [ x ] ∈ B ( M ).[ x ] ∈ { [ p ] } ⇔ every closed subset of B ( M ) that contains [ p ] also con-tains [ x ] ⇔ there exists no closed subset of B ( M ) that contains [ p ]that does not contain [ x ] ⇔ there exists no open neighbourhood of [ x ] in M thatdoes not contain [ p ] ⇔ every open neighbourhood of [ x ] in M contains [ p ] ⇔ [ x ] ⊲ [ p ] (by proposition 32)Thus { [ p ] } = { [ p ] } ∪ { [ x ] : [ x ] ∈ B ( M ) , [ x ] ⊲ [ p ] } . (cid:3) We may now more readily consider the question of whether or not thesingleton abstract boundary point sets { [ p ] } are open or closed in the stronglyattached point topology on M . Proposition 34
The singleton abstract boundary point sets { [ p ] } are notopen in the strongly attached point topology on M . They are also notclosed in the strongly attached point topology on M if and only if thereexists [ q ] ∈ B ( M ), [ q ] = [ p ], such that [ q ] ⊲ [ p ]. Proof:
Consider an abstract boundary point [ p ]. From corollary 29, any24pen set in M that contains [ p ] will necessarily have non-empty intersectionwith M , and thus { [ p ] } is not an open set.By corollary 33, the closure in M of an abstract boundary point [ p ] is { [ p ] } = { [ p ] } ∪ { [ x ] : [ x ] ∈ B ( M ) , [ x ] ⊲ [ p ] } . If there exists a [ q ] such that[ q ] ⊲ [ p ], [ q ] = [ p ], then by corollary 33, { [ p ] } contains at least [ p ] and [ q ],and therefore { [ p ] } is not closed. Similarly, if { [ p ] } is not closed, then { [ p ] } contains at least 2 elements [ p ] and [ q ] such that [ p ] = [ q ] and [ q ] ⊲ [ p ]. Itfollows that { [ p ] } is not closed in the strongly attached point topology on M if and only if there exists [ q ] ∈ B ( M ), [ q ] = [ p ], such that [ q ] ⊲ [ p ]. (cid:3) Proposition 35
The open sets of the induced topology on B ( M ) ⊂ M ,where M has the strongly attached point topology, are arbitrary unions ofthe B i sets defined in the basis W . Proof:
Let T M be the strongly attached point topology on M . Thesubspace topology on B ( M ) is the collection of sets T B ( M ) = { U ∩ B ( M ) : U ∈ T M } . The topology T M is the collection of arbitrary unions of the U i ∪ B i sets of the basis W . The intersection of these sets with B ( M ) is thereforethe collection of arbitrary unions of the B i sets. (cid:3) M to M We now consider the inclusion map i : M → M = M ∪ B ( M ) | i ( p ) = p . Asin the case of the attached point topology, it can be shown that the inclusionmap is an embedding. Proposition 36 If M has the strongly attached point topology, then theinclusion mapping i : M → M| i ( p ) = p is an embedding.25 roof: The inclusion mapping i is an embedding if it is a homeomor-phism of M onto i ( M ) in the subspace topology on i ( M ) ∩ M . Clearly i is a bijection of M onto i ( M ). Now let T M be the usual topology on M consisting of the collection of open sets { U i } , T M the strongly attached pointtopology on M as defined by the basis elements of W , i.e., T M is the collec-tion of arbitrary unions of the U i ∪ B i sets, and T i ( M ) the subspace topologyon i ( M ) ∩ M . The subspace topology T i ( M ) is therefore the collection ofsets T i ( M ) = { U k } . Clearly both i and i − are continuous with respect to T M and T i ( M ) . It has thus been demonstrated that i : M → M | i ( p ) = p is ahomeomorphism onto its image in the induced topology and is therefore anembedding. (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. Proposition 37
For the inclusion mapping i : M → M| i ( p ) = p , i ( M ) isopen and not closed in the strongly attached point topology on M , i ( M ) = M and ∂ ( i ( M )) = B ( M ). Proof:
Since i ( M ) = M , it follows from proposition 26 and proposition27 that i ( M ) is open and not closed in the strongly attached point topol-ogy on M . Because i ( M ) is open, ∂ ( i ( M )) = ∂ ( M ) = { x ∈ M\M :every open neighbourhood of x has non-empty intersection with M} . Con-sider an abstract boundary point [ p ] ∈ B ( M ) = M\M . From corollary 29,26very open neighbourhood of [ p ] has non-empty intersection with M , and so[ p ] ∈ ∂ ( i ( M )). Thus ∂ ( i ( M )) = B ( M ). Now i ( M ) = i ( M ) ∪ ∂ ( i ( M )) = M ∪ B ( M ) = M . (cid:3) A number of important properties of the strongly attached point topologywill now be presented.Due to the way that abstract boundary points are constructed, two ab-stract boundary points may share some of the same topological information.For example, if [ p ] = [ q ] then any envelopment that produces a boundary setbelonging to [ p ] will also produce a boundary set belonging to [ q ] and viceversa. Likewise, in the case that [ p ] covers [ q ] we have that [ p ] contains [ q ]in some sense. When [ p ] and [ q ] are realised as boundary sets A ⊆ ∂ ( φ ( M ))and B ⊆ ∂ ( φ ′ ( M )) respectively, the topological structure of φ ( M ) near A incorporates the topological structure of φ ′ ( M ) near B . In this way, whenwe consider the abstract boundary point [ p ] relative to M , we are also con-sidering the abstract boundary point [ q ]. Alternatively, we may have the casewhere [ p ] and [ q ] are not in contact at all, and are somehow ‘separate’ fromeach other.A topology on M should therefore be descriptive of the topological ‘con-tact’ properties between abstract boundary points. It can be seen that thestrongly attached point topology describes the separation properties of ab-stract boundary points in a natural way in that greater levels of separationbetween abstract boundary points with respect to the covering relation cor-27espond to greater levels of separation with respect to the usual topologicalseparation axioms.We begin by defining what it means for an abstract boundary point to bein contact with another abstract boundary point. In some sense the contactrelation is a weaker form of the covering relation. If [ p ] and [ q ] are in contact,then they contain some of the same topological information, but not as muchas if [ p ] covered [ q ] or [ q ] covered [ p ]. Definition 38 (Contact ⊥ ) Let p ∈ ∂ ( φ ( M )) and q ∈ ∂ ( φ ′ ( M )) be twoenveloped boundary points of M . They are said to be in contact (denoted p ⊥ q ) if for all open neighbourhoods U and V of p and q respectively, U ⊓ V := φ − ( U ∩ φ ( M )) ∩ φ ′− ( V ∩ φ ′ ( M )) = ∅ . Definition 39 (Contact ⊥ (sequence definition)) Two boundary points p ∈ ∂ ( φ ( M )) and q ∈ ∂ ( φ ′ ( M )) are in contact (denoted p ⊥ q ) if there exists asequence { p i } ⊂ M such that { φ ( p i ) } has p as an endpoint and { φ ′ ( p i ) } has q as an endpoint.Definitions 38 and 39 are equivalent. For a proof of this see lemma 6.3 of[14]. Definition 40 (Abstract boundary points in contact) Two abstract bound-ary points [ p ] and [ q ] are in contact , denoted [ p ] ⊥ [ q ], if p ⊥ q for boundarypoint representatives p and q .This definition can be shown to be well-defined. See theorem 3.10 of [14]. Definition 41 (Separation of boundary points k ) Two boundary points p ∈ ( φ ( M )) and q ∈ ∂ ( φ ′ ( M )) are separate (denoted p k q ) if there is nosequence { p i } ⊂ M for which { φ ( p i ) } → p and { φ ′ ( p i ) } → q . Equivalently,the boundary points p and q are separate if there exist open neighbourhoods U and V of p and q respectively such that φ − ( U ∩ φ ( M )) ∩ φ ′− ( V ∩ φ ′ ( M )) = ∅ . Equivalently, from definition 39, two boundary points p ∈ ∂ ( φ ( M )) and q ∈ ∂ ( φ ′ ( M )) are separate if they are not in contact. Definition 42 (Separation of abstract boundary points) Two abstract bound-ary points [ p ] and [ q ] are separate , denoted [ p ] k [ q ], if p k q for boundarypoint representatives p and q . Equivalently, [ p ] k [ q ] if they are not in contact.Similar to definition 40, this definition can be shown to be well-defined.See theorem 3.10 of [14].The results which follow relate to M with the strongly attached pointtopology. Proposition 43
Two abstract boundary points [ p ] and [ q ] are T separable,i.e., they are Hausdorff separable, if and only if [ p ] k [ q ]. Proof: ( ⇐ ) If [ p ] k [ q ], then [ p ] and [ q ] are T separable.If [ p ] k [ q ] then there exists an open neighbourhood U of p ∈ ∂ ( φ ( M ))and an open neighbourhood V of q ∈ ∂ ( φ ′ ( M )) such that φ − ( U ∩ φ ( M )) ∩ φ ′− ( V ∩ φ ′ ( M )) = ∅ . We also have that p is strongly attached to φ − ( U ∩ φ ( M )), q is strongly attached to φ ′− ( V ∩ φ ′ ( M )), and from lemma 13, p is not strongly attached to φ ′− ( V ∩ φ ′ ( M )) and q is not strongly at-29ached to φ − ( U ∩ φ ( M )). Furthermore, it also follows from lemma 13 thatthere exist no abstract boundary points which are strongly attached to both φ − ( U ∩ φ ( M )) and φ ′− ( V ∩ φ ′ ( M )). There therefore exists an open neigh-bourhood of [ p ], φ − ( U ∩ φ ( M )) ∪ B U , [ p ] ∈ B U , and an open neighbourhoodof [ q ], φ ′− ( V ∩ φ ′ ( M )) ∪ B V , [ q ] ∈ B V , such that their intersection is empty.The abstract boundary points [ p ] and [ q ] are therefore Hausdorff separated.( ⇒ ) If [ p ] and [ q ] are T separable, then [ p ] k [ q ].If [ p ] and [ q ] are Hausdorff separated then there exist open neighbour-hoods U p of [ p ] and U q of [ q ] in the strongly attached point topology suchthat U p ∩ U q = ∅ . Since [ p ] is contained in U p , by corollary 30, [ p ] is stronglyattached to the open set V p = U p ∩ M . Now, since [ p ] is strongly attachedto V p , there exists an open neighbourhood W p in c M of p ∈ ∂ ( φ ( M )) suchthat W p ∩ φ ( M ) ⊆ φ ( V p ). Similarly, there exists an open neighbourhood W q in c M ′ of q ∈ ∂ ( φ ′ ( M )) such that W q ∩ φ ′ ( M ) ⊆ φ ′ ( V q ), where V q isthe open set V q = U q ∩ M in M . Now, since W p ∩ φ ( M ) ⊆ φ ( V p ) and W q ∩ φ ′ ( M ) ⊆ φ ′ ( V q ), and V p ⊆ U p and V q ⊆ U q , where U p ∩ U q = ∅ , itfollows that φ − ( W p ∩ φ ( M )) ∩ φ ′− ( W q ∩ φ ′ ( M )) = ∅ , and so [ p ] k [ q ]. (cid:3) Proposition 44
Two abstract boundary points [ p ] and [ q ] are T separatedif and only if [ p ] ⋫ [ q ] and [ q ] ⋫ [ p ]. Proof: ( ⇐ ) If [ p ] ⋫ [ q ] and [ q ] ⋫ [ p ], then [ p ] and [ q ] are T separated.If [ p ] ⋫ [ q ] and [ q ] ⋫ [ p ], then by corollary 32 there exists an open neigh-bourhood N p of [ p ] and an open neighbourhood N q of [ q ] such that [ q ] / ∈ N p p ] / ∈ N q . This is a statement of the T separation axiom.( ⇒ ) If [ p ] and [ q ] are T separated, then [ p ] ⋫ [ q ] and [ q ] ⋫ [ p ].If [ p ] and [ q ] are T separated, then there exists an open neighbourhood N p of [ p ] and an open neighbourhood N q of [ q ] such that [ q ] / ∈ N p and [ p ] / ∈ N q .It then follows directly from corollary 32, that [ p ] ⋫ [ q ] and [ q ] ⋫ [ p ]. (cid:3) Proposition 45
Two abstract boundary points [ p ] and [ q ] are T separatedif and only if [ p ] ⋫ [ q ] or [ q ] ⋫ [ p ]. Proof: ( ⇐ ) If [ p ] ⋫ [ q ] or [ q ] ⋫ [ p ], then [ p ] and [ q ] are T separated.By corollary 32, if [ q ] ⋫ [ p ], then there exists an open neighbourhood N q of [ q ] such that [ p ] / ∈ N q , and so [ p ] and [ q ] are T separated. Likewise, if[ p ] ⋫ [ q ], then [ p ] and [ q ] are T separated.( ⇒ ) If [ p ] and [ q ] are T separated, then [ p ] ⋫ [ q ] or [ q ] ⋫ [ p ].If [ p ] and [ q ] are T separated, then there exists an open neighbourhood N p of [ p ] such that [ q ] / ∈ N p , or there exists an open neighbourhood N q of [ q ]such that [ p ] / ∈ N q . If [ p ] / ∈ N q , then by corollary 32, [ q ] ⋫ [ p ]. Likewise, if q / ∈ N p , then [ p ] ⋫ [ q ]. (cid:3) The results of this section are summarised in table 1 which shows thecorrespondence between the contact properties of two enveloped boundarypoints p ∈ ∂ ( φ ( M )) and q ∈ ∂ ( φ ′ ( M )) and the topological relationship of31he respective abstract boundary points [ p ] and [ q ] in M with the stronglyattached point topology. We provide examples of the second and third rela-tionships in figures 9 and 10, respectively.Relationship between envelopedboundary points p ∈ ∂ ( φ ( M )) and q ∈ ∂ ( φ ′ ( M )) Topological relationship of the abstractboundary points [ p ] and [ q ] p ∼ q [ p ] = [ q ] p ⊲ q , q ⋫ p or q ⊲ p , p ⋫ q [ p ] and [ q ] are T separated (proposition45)[ p ] and [ q ] are not T separated (propo-sition 44) p ⊥ q , p ⋫ q , q ⋫ p [ p ] and [ q ] are T separated (proposition44)[ p ] and [ q ] are not T separated (propo-sition 43) p k q [ p ] and [ q ] are T separated (proposition43)Table 1: The left hand column shows the possible relationships betweenboundary points p and q of envelopments φ and φ ′ , respectively, of M ; theright hand column shows the corresponding topological relationships betweenthe associated abstract boundary points [ p ] and [ q ] in M with the stronglyattached point topology.Hausdorff separability is lost between abstract boundary points whichare in contact with each other, and therefore, also when one of the abstractboundary points covers the other. In many ways, this is an expected result.As has been stated previously, two abstract boundary points which are incontact with one another share a certain amount of topological information,and thus they do not represent two truly distinct points. This property is32Sfrag replacements R R p Bφ ( M ) φ ′ ( M ) q Figure 9: the boundary point p ∈ ∂ ( φ ( M )) is equivalent to the closed bound-ary set B ⊂ ∂ ( φ ′ ( M )), where q ∈ B . It follows that p ⊲ q , but q ⋫ p .reflected in the loss of Hausdorff separation in the strongly attached pointtopology. And so, while it is desirable that a topology for M be Hausdorff,it can be seen that the lack of separation between abstract boundary pointsactually provides us with information about the structure of the abstractboundary itself. Moreover, it can be argued that Hausdorff separation is notlost between truly distinct abstract boundary points (namely those whichare separate). Instead, it is lost between abstract boundary points whichrepresent different parts of some ‘larger’ entity.We note that, in general, the strongly attached point topology on M willbe T separated only, as there will be occurrences of p ⊲ q , q ⋫ p for boundarypoints p ∈ ∂ ( φ ( M )) and q ∈ ∂ ( φ ′ ( M )).We will now determine if the strongly attached point topology is first33Sfrag replacements φ ′ ( M ) c M ′ φ ′ ( λ ) φ ′ ( λ ) qφ ( M ) c M φ ( λ ) φ ( λ ) pt ψ t ′ ψ ′ ∂ ( φ ( M )) ∂ ( φ ′ ( M ))Figure 10: two envelopments of the two-dimensional Misner space-time withrespective metrics: ds = 2 dψdt + t ( dψ ) and ds = − dψ ′ dt ′ + t ′ ( dψ ′ ) . Thecurves λ and λ are null geodesics. We may construct a sequence along φ ( λ ) that converges to p . It follows that the image of this sequence under φ ′ converges to q . The boundary points p and q are therefore in contact.The curve φ ′ ( λ ) is an element of a class of geodesics that spiral around thespace-time and approach the waist. The image under φ of each such geodesicis a straight vertical line similar to φ ( λ ) that approaches some point of theboundary set ∂ ( φ ( M )) of which p is an element. It follows that p ⋫ q aswe can construct a sequence that converges to q along one of the spiralinggeodesics in φ ′ ( M ) whose image under φ does not converge to p . By a similarargument it can be shown that q ⋫ p .34ountable. Proposition 46
The strongly attached point topology on M is first count-able. 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 with the strongly attached point topology, we firstly considerthe case where x ∈ M . Given the existence of a complete metric d on M ,we know from the proof of proposition 26, that for n ∈ N , the open balls U n = { p ∈ M : d ( x, p ) < /n } based at the point x have no attached ab-stract boundary points and therefore no strongly attached abstract boundarypoints. The sets U n ∪ B n = U n are basis elements of W , and so U , U ,... isa sequence of open neighbourhoods of x .Let V be an open neighbourhood of x . Thus V is an arbitrary union ofbasis elements A i which implies that x ∈ A k = U k ∪ B k ⊆ V for some A k inthe union. It is possible to choose an n ∈ N , such that, for the open ball U n , U n ⊂ U k . Thus U n ⊆ V . We have therefore shown that M is first countableat x , for all x ∈ M .Now we consider an abstract boundary point [ p ] ∈ B ( M ), where p is aboundary point of some envelopment ( M , c M , φ ). Similarly to before, giventhe existence of a complete metric d on c M , we can define a series of open ballsof p in c M by O n = { y ∈ c M : d ( p, y ) < /n } , n ∈ N . We therefore have aseries of sets in M that contain [ p ] defined by [ φ − ( O n ∩ φ ( M ))] ∪ B O n , wherethe B O n are the collections of abstract boundary points that are strongly35ttached to φ − ( O n ∩ φ ( M )). Clearly these sets are open in the stronglyattached point topology on M as they are elements of the collection of basissets W . Every open set V in M that contains [ p ] is an arbitrary union of A i = U i ∪ B i sets. One of the B j sets therefore contains [ p ], and so [ p ] isstrongly attached to U j . There thus exists an open neighbourhood N of p in c M such that N ∩ φ ( M ) ⊆ φ ( U j ). Now, there exists an n ∈ N , such that O n ⊂ N , and so [ φ − ( O n ∩ φ ( M ))] ∪ B O n ⊆ V . This means that M is firstcountable at [ p ], for all [ p ] ∈ B ( M ).We have thereby shown that the strongly attached point topology for M is first countable. (cid:3) When presented with a solution to the Einstein field equations in a partic-ular coordinate system, it is not necessarily the case that these coordinatesproperly display all of its global and physical properties. In practice, thisoften amounts to determining if the space-time is a proper subset of another,larger space-time. The abstract boundary is therefore the natural bound-ary construction to use when considering extensions to space-times, given itsutility in dealing with multiple envelopments at once. An envelopment inwhich all of the global features of a space-time are evident may therefore bereferred to as an optimal embedding.In order to be able to choose an envelopment in which all of the globalfeatures of a space-time are properly displayed, the structure of the abstractboundary must be understood. If a boundary set of an abstract boundary36oint is present in an envelopment, then we would like to know how the ab-stract boundary point represented by this boundary set is related to otherabstract boundary points. More specifically, we seek to know things like:is the abstract boundary point represented by that boundary set containedin some other abstract boundary point in some sense, i.e., is the abstractboundary point covered by some other abstract boundary point? And there-fore, is there a better, more complete way of displaying the boundary ofthe space-time in an envelopment? If there exists an envelopment in whichmore topological and physical information can be displayed, then clearlywe should choose that envelopment. Understanding the contact propertiesbetween abstract boundary points is therefore essential when considering op-timal embeddings.The contact properties that were defined earlier (definition 40 and defi-nition 42) may be used to define subsets of B ( M ) referred to as partial crosssections. Partial cross sections provide us with a way of abstracting the ideaof envelopments as pictures of the boundary. The abstract boundary is avery large object. In some sense, the complete abstract boundary of a man-ifold M contains too much information. As discussed previously, differentabstract boundary points can share large amounts of the same topological in-formation. It is therefore not necessary to consider every abstract boundarypoint in order to understand the structure of the abstract boundary. A par-tial cross section is a ‘slice’ through the abstract boundary containing onlyabstract boundary points which are topologically distinct from each other.Partial cross sections are therefore important because, ideally, they can beused to simplify the abstract boundary to something more manageable. Inturn this can lead to the realisation of optimal embeddings. For furtherdetails on optimal embeddings see [14].37 efinition 47 (Partial cross section σ ) Let σ ⊂ B ( M ). σ is a partial crosssection if for every [ p ], [ q ] ∈ σ , [ p ] k [ q ] or [ p ] = [ q ].Of particular interest are partial cross sections of the following form: Example 48
Each envelopment ( M , c M , φ ) defines a partial cross section σ φ := { [ p ] | p ∈ ∂ ( φ ( M )) } . These σ φ sets are important because we know what the topology on thesesets should look like. Each abstract boundary point in σ φ has a boundarypoint representative in the topological boundary ∂ ( φ ( M )). The topology ofthis set is well defined by the topology on c M and agrees with the relativetopology on φ ( M ), and hence it also agrees with the topology on M byvirtue of the embedding φ . Each σ φ therefore has a natural topology definedon it by the given envelopment ( M , c M , φ ). Definition 49
Let φ : M → c M be an envelopment, and σ φ the partialcross section induced by φ . A natural topology T σ φ is defined upon σ φ bythe topology of c M . Let N be an open neighbourhood of c M . We then takea set U to be an open set of σ φ ( U ∈ T σ φ ) if and only if U = { [ p ] ∈ σ φ :the singleton representative boundary point p ∈ ∂ ( φ ( M )) is an element of N ∩ ∂ ( φ ( M )) } for some open neighbourhood N .As mentioned previously, the topology on ∂ ( φ ( M )) is that induced by thetopology on c M . Because the elements of σ φ and ∂ ( φ ( M )) are in one-to-onecorrespondence with each other, it follows that the collection T σ φ of open setsof σ φ given by definition 49 is indeed a topology on σ φ .38 emma 50 Let φ : M → c M be an envelopment, and σ φ the partial crosssection induced by φ . The topological space ( σ φ , T σ φ ) is Hausdorff. Proof:
Let [ p ], [ q ] ∈ σ φ , [ p ] = [ q ], where p and q are distinct boundarypoints of ∂ ( φ ( M )). Since the topology of c M is Hausdorff, there exist dis-joint open neighbourhoods U and V of p and q , respectively, in c M . Nowif we define U ∗ = φ − ( U ∩ φ ( M )) and V ∗ = φ − ( V ∩ φ ( M )), it followsthat U ∗ ∩ V ∗ = ∅ , and [ p ] is strongly attached to U ∗ and [ q ] is strongly at-tached to V ∗ . Define A U ∗ = U ∗ ∪ B U ∗ and A V ∗ = V ∗ ∪ B V ∗ , where B U ∗ isthe set of all abstract boundary points in σ φ which are strongly attached to U ∗ (so [ p ] ∈ B U ∗ ) and B V ∗ is the set of all abstract boundary points in σ φ which are strongly attached to V ∗ (so [ q ] ∈ B V ∗ ). Thus A U ∗ ∩ σ φ = B U ∗ and A V ∗ ∩ σ φ = B V ∗ are open sets of T σ φ and open neighbourhoods of [ p ] and [ q ] re-spectively. Consider some [ r ] ∈ σ φ , where r ∈ ∂ ( φ ( M )) and r = p , such that[ r ] ∈ B V ∗ . There therefore exists an open neighbourhood W of r in c M suchthat W ∩ φ ( M ) ⊆ φ ( V ∗ ). Now assume that [ r ] ∈ B U ∗ . This implies that thereexists an open neighbourhood X of r in c M such that X ∩ φ ( M ) ⊆ φ ( U ∗ ).Since U ∗ ∩ V ∗ = ∅ and W ∩ φ ( M ) ⊆ φ ( V ∗ ) and X ∩ φ ( M ) ⊆ φ ( U ∗ ), itfollows that [ X ∩ φ ( M )] ∩ [ W ∩ φ ( M )] = ∅ . We also have that X and W areboth open neighbourhoods of r , and so [ X ∩ φ ( M )] ∩ [ W ∩ φ ( M )] = ∅ . Wetherefore have a contradiction. This implies that B U ∗ and B V ∗ are disjointopen neighbourhoods of [ p ] and [ q ] respectively, thereby demonstrating thatthe topological space ( σ φ , T σ φ ) is Hausdorff. (cid:3) In practice, the abstract boundary of a space-time is studied by con-sidering its envelopments. It is therefore highly desirable that the naturaltopology T σ φ of a partial cross section σ φ agrees with the topology on σ φ B ( M ).In the following proposition we show, assuming a condition holds, thatthe natural topology T σ φ of a partial cross section σ φ agrees with the topologyon σ φ induced by the strongly attached point topology. Condition 51
Consider an envelopment ( M , c M , φ ) with boundary ∂ ( φ ( M )) = ∅ . There exists an open neighbourhood V of ∂ ( φ ( M )) in c M and a C con-gruence of curves { λ p } on V such that:1. λ p passes through p ∈ ∂ ( φ ( M )) from one side to the other side of ∂ ( φ ( M )) where it exists as a surface2. λ p ∩ ∂ ( φ ( M )) = { p } { λ p } is non-intersecting4. For each p , λ p : ( − α, β ) → V , where α, β ∈ R + , such that: λ p (0) = p , λ p ( − α ), λ p ( β ) ∈ V \ V , λ p ( − α, ⊂ φ ( M ), λ p (0 , β ) ⊂ c M\ φ ( M ) or λ p (0 , β ) ⊂ φ ( M ).See figure 11. The possibility that λ p (0 , β ) ⊂ φ ( M ) is included in (iv) tocover the case where, for a sufficiently small open neighbourhood U of p in c M , U \ ∂ ( φ ( M )) ⊆ φ ( M ). Proposition 52
Let T σ φ be the topology on σ φ , defined by the topology of c M , given in definition 49, and let T σ φ ( str ) be the topology on σ φ induced by40Sfrag replacements φ ( M ) V congruence of curves { λ p } c M ∂ ( φ ( M ))Figure 11: an example of an envelopment ( M , c M , φ ) which satisfies condition51.the strongly attached point topology on M . If ( M , c M , φ ) obeys condition51, then T σ φ = T σ φ ( str ) . Proof:
1) If U is an open set of T σ φ , then U is an open set of T σ φ ( str ) .Let U be an open set of T σ φ . This means that for some open set N of c M such that N ∩ ∂ ( φ ( M )) = ∅ , U = { [ p ] ∈ σ φ : p ∈ N ∩ ∂ ( φ ( M )) } . It is clearthat for each p ∈ N ∩ ∂ ( φ ( M )), [ p ] is strongly attached to φ − ( N ∩ φ ( M ))We now consider whether any other abstract boundary points in σ φ arestrongly attached to φ − ( N ∩ φ ( M )). Suppose q ∈ ∂ ( φ ( M )) \ N ∩ ∂ ( φ ( M )).Thus q ∈ c M\ N which is an open set disjoint from the open set N . Every41pen neighbourhood of q will have non-empty intersection with c M\ N ∩ φ ( M )and so [ q ] is not strongly attached to φ − ( N ∩ φ ( M )).Now suppose that q ∈ N ∩ ∂ ( φ ( M )) \ N ∩ ∂ ( φ ( M )). The envelopment( M , c M , φ ) obeys condition 51, and so there exists an open neighbourhood V of ∂ ( φ ( M )) in c M which satisfies condition 51. Consider the set Y = S p λ p where p ∈ N ∩ ∂ ( φ ( M )) \ N ∩ ∂ ( φ ( M )). Now define the set N ∗ = c M\ Y ∩ N ∩ φ ( M ). Clearly N ∗ is an open subset of φ ( M ). Consider the point q ∈ N ∩ ∂ ( φ ( M )). Since N is an open set, there exists a small open neigh-bourhood N q of q in N such that N q ∩ ∂ ( φ ( M )) ⊆ N . By condition 51, a smallopen ball B ǫ of radius ǫ about q can be chosen such that B ǫ ⊆ N q and no curve λ p enters B ǫ , where p ∈ N ∩ ∂ ( φ ( M )) \ N ∩ ∂ ( φ ( M )). Now B ǫ ∩ φ ( M ) ⊆ N ∗ and thus N ∗ is non-empty. It follows that [ q ] is strongly attached to φ − ( N ∗ )for all q ∈ N ∩ ∂ ( φ ( M )). If q ∈ N ∩ ∂ ( φ ( M )) \ N ∩ ∂ ( φ ( M )), the curve λ q enters every open neighbourhood of q , and so [ q ] is not strongly attached to φ − ( N ∗ ). If q ∈ ∂ ( φ ( M )) \ N ∩ ∂ ( φ ( M )) we know that [ q ] is not stronglyattached to φ − ( N ∩ φ ( M )) and since N ∗ ⊆ N ∩ φ ( M ), [ q ] is not stronglyattached to φ − ( N ∗ ). Thus U is the open set of all abstract boundary pointsin σ φ which are strongly attached to the open set φ − ( N ∗ ) of M and so U isan open set of T σ φ ( str ) .2) If U is an open set of T σ φ ( str ) , then U is an open set of T σ φ .Let U be an open set of T σ φ ( str ) . There therefore exists an open set S i A i = S i U i ∪ B i of M , where each U i is a non-empty open set of M and B i is the set of all abstract boundary points which are strongly attached to U i , such that U = ( S i U i ∪ B i ) ∩ σ φ = ( S i B i ) ∩ σ φ .Consider [ p ] ∈ ( S i B i ) ∩ σ φ , where p ∈ ∂ ( φ ( M )). There exists a B i such42hat [ p ] ∈ B i and thus [ p ] is strongly attached to U i . This means that thereexists an open neighbourhood N p of p in c M such that N p ∩ φ ( M ) ⊆ φ ( U i ).Consider a boundary point q ∈ ∂ ( φ ( M )) such that q ∈ N p . It is clear that[ q ] ∈ B i and thus [ q ] ∈ U . The set W = S [ p ] ∈ U N p is an open set in c M .Consider the open set in T σ φ defined by A = { [ p ] : p ∈ W ∩ ∂ ( φ ( M )) } . It isclear that U ⊆ A and, from the above argument, that A ⊆ U . Thus U is anopen set of T σ φ . (cid:3) Direction 2 of the previous proof is quite straightforward. Direction 1on the other hand, is more complicated and requires us to invoke condition51. We have to use this condition due to the existence of boundary points p ∈ ∂ ( φ ( M )) that are strongly attached to φ − ( N ∩ φ ( M )) but are notelements of N . The existence of these boundary points makes it difficult toconstruct an open neighbourhood of T σ φ ( str ) that doesn’t contain abstractboundary points additional to those contained in U . Even so, condition 51is not very restrictive and may even hold in general. At the least, we havebeen unable to construct a space-time in which it does not hold. There are many topologies that can be placed on M . They will not all bephysically useful, however. Ultimately, a topology should provide a structurefor M which aids us in answering physical questions about M . Ideally then,the topology should connect the abstract boundary to the manifold in aphysically meaningful way, and the resulting structure on M should conformto many of our intuitive ideas regarding the behaviour of ‘missing points’,i.e., abstract boundary points, from the manifold M .43he strongly attached point topology was defined similarly to the at-tached point topology but with one important difference. This difference,related to the way in which abstract boundary points are ‘attached’ to opensets of M , means that the strongly attached point topology does not need toinclude collections of abstract boundary points. In the attached point topol-ogy these sets were necessary to ensure that the basis for the topology waswell-defined. In some sense, because the abstract boundary points are morefirmly connected to the manifold in the strongly attached point topology,we avoid having to add more open sets to the topology. It is interesting tonote that as a consequence of this, every open neighbourhood of an abstractboundary point in the strongly attached point topology necessarily containssome part of the manifold M , thereby encapsulating the true essence of aboundary point.Another consequence of the strongly attached point topology not con-taining collections of abstract boundary points is that there exist abstractboundary points which are not Hausdorff separated from each other. While ithas been argued that a physically useful topology for a space-time should beHausdorff [16], the lack of Hausdorff separation between abstract boundarypoints in the strongly attached point topology, nevertheless, contains use-ful information about the boundary. It was demonstrated that two abstractboundary points are Hausdorff separable if and only if they are not in contact.Intuitively, this makes sense as two abstract boundary points which are incontact share much of the same topological information, and therefore theydo not represent two points which are distinct from each other. It thereforeseems reasonable that abstract boundary points which are in contact witheach other cannot be separated by disjoint open sets. It is also worth notingthat there is a natural relationship between the separation axioms that two44bstract boundary points obey, and how much topological information theyshare. As propositions 43 through 45 show, as two abstract boundary pointsshare more of the same topological information, they obey fewer separationaxioms. For example, two abstract boundary points which are in contactare T separable, but not T separable, while if one abstract boundary pointcovers the other, they are T separable, but not T separable. Therefore,while separation is lost between abstract boundary points, it is lost in a waydirectly related to the amount of overlap between the abstract boundarypoints.The strongly attached point topology possesses a number of other inter-esting properties which suggest that it is an appropriate topology for M .One such property is that the description of M and B ( M ) in the stronglyattached point topology agrees with many of our intuitive ideas about thenature of a space and its boundary. Traditionally, singularities are typicallyviewed as ‘points’ missing from a space-time. We can approach these missingpoints from within the space-time, becoming arbitrarily close to them, butwe cannot reach them. In some sense then, these missing points make upthe ‘closure’ of the space-time, and ideally, a topology on M should reflectthis. The strongly attached point topology agrees with this notion in thesense that M is open and not closed, B ( M ) is closed and not open, and M can be embedded identically into M . The strongly attached point topologytherefore provides a natural way of viewing the structure of M in that itcan be seen as M with the addition of a topological boundary made up ofabstract boundary points.Perhaps the most important property of the strongly attached pointtopology is that the topology induced by the strongly attached point topol-ogy on a partial cross section σ φ associated with an envelopment φ : M → c M σ φ . In practice, the abstractboundary is studied via envelopments of the manifold M . Consequently, theembedded manifold φ ( M ) and its topological boundary ∂ ( φ ( M )) alreadyhave a topology defined on them with which it is very easy to work. It istherefore very useful that the topologies agree on the partial cross sections σ φ as it means that any topological result which holds in an envelopment(which, again, is where the abstract boundary is studied in practice) willalso hold with respect to the larger topology on B ( M ).Without a topology on M we cannot say ‘where’ singular points are withrespect to the manifold M . A topology on M should therefore relate theabstract boundary back to the manifold M . Moreover, it should ideally doso in a natural way, i.e., the topology should describe the singular pointsin a way that agrees with our intuitive ideas of how a singularity is relatedto the manifold. It has been shown that the strongly attached point topol-ogy does indeed relate the abstract boundary back to the manifold in a waythat encompasses many of our intuitive notions of the nature of a topologicalboundary. For these reasons, the strongly attached point topology appears tobe a particularly good choice for a topology on the set comprising a manifoldand its abstract boundary. Acknowledgements
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