A generalized manifold topology for branching space-times A Generalized Manifold Topology for Branching Space-Times Thomas Müller*y The logical theory of branching space times, which provides a relativistic framework for studying objective indeterminism, remains mostly disconnected from discussions of space time theories in philosophy of physics. Earman has criticized the branching ap proach and suggested “pruning some branches from branching space time.” This article identifies the different order theoretic versus topological perspective of both discus sions as a reason for certain misunderstandings and tries to remove them. Most impor tant, we give a novel, topological criterion of modal consistency that usefully general izes an earlier criterion, and we introduce a differential geometrical version of branching space times as a non Hausdorff ðgeneralizedÞ manifold. 1. Introduction. Discussions of determinism and indeterminism play an important role in many areas of philosophy. In philosophy of science, they have acted as probes into the basic structure of many physical theories; in philosophical logic, such discussions have triggered the development of a number of different logical systems, some of which have found application in computer science. When it comes to defining indeterminism, one can discern two related but technically different approaches. The basic idea behind indeterminism is that given the way things are at one moment in time, more than one future course of events is possible. One way to spell this out is to start with a class of separate possible courses of events and the notion of a state at a moment. If the same state occurs in at least two different courses of events that dis- *To contact the author, please write to: Department of Philosophy, Utrecht University, Jans kerkhof 13a, 3512 BL Utrecht, The Netherlands; e mail: Thomas.Mueller@uni konstanz.de. yResearch leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme ðFP7/2007 2013Þ/ERC grant agreement 263227 and from the Dutch Organization for Scientific Research, grant NWO VIDI 276 20 013. Thanks to Nuel Belnap and Tomasz Placek for many stimulating discussions. Philosophy of Science, 80 (December 2013) pp. 1089 1100. 0031-8248/2013/8005-0030$10.00 Copyright 2013 by the Philosophy of Science Association. All rights reserved. 1089 http://nbn-resolving.de/urn:nbn:de:bsz:352-268027 agree about the future development after the respective occurrences, the class is indeterministic. If that class is given via a scientific theory, that the- ory is accordingly diagnosed to be indeterministic. As a way of classifying scientific theories, this approach was pioneered by Montague ð1962Þ. After several refinements ðsee, e.g., Earman 2006Þ, many of which were triggered by specific issues of space-time theories, this approach grounds the gener- ally accepted definition of indeterminism for scientific theories: “a theory is deterministic if, and only if: for any two of its models, if they have instan- taneous slices that are isomorphic, then the corresponding final segments are also isomorphic” ðButterfield 2005Þ. A different way to model indeter- minism is to start not with separate courses of events ðcalled “models” in the above quote, but see sec. 2Þ to be matched via their states at moments but with a unified structure of moments within which one can identify the pos- sible courses of events—histories—as substructures. This approach is more in line with an indexical understanding of branching future possibilities, ac- cording to which different possible courses of events can literally share a past segment. Such branching history structures were developed in Prior’s tense logic ðPrior 1967Þ; an important early paper is Thomason ð1970Þ. In so- called branching time ðBT; a somewhat misleading label, as time does not branch, only temporal histories doÞ, a single history is pictured as a line- arly ordered set of moments carved from a global partial ordering of mo- ments. BT is used as a background for the “seeing to it that” logic of agency ðBelnap, Perloff, and Xu 2001Þ and arguably corresponds well to the phe- nomenology of an open future. However, the framework is nonrelativistic: a moment ðan element of the orderingÞ has to represent all of space simul- taneously. Theories of branching space-times ðBSTÞ extend the branching histo- ries idea by taking histories to be space-times rather than linearly ordered chains of moments. Belnap’s version of BST ðBST92; Belnap 1992Þ pro- vides a mathematically rigorous theory of objective indeterminism in a rel- ativistic setting. BST92 retains BT’s underlying algebraic approach of start- ing with a partially ordered set that can be called Our World as it represents all events that are, were, or will be possible. The notion of modal consis- tency, that is, possible co-occurrence within a single consistent course of events, which in the case of BT comes down to order relatedness ðlinearityÞ, has to be extended: in the case of BST92, consistency is taken to mean the existence of a common upper bound ðtwo events are compatible if there is, so to speak, a perspective from which both events have occurredÞ. Build- ing on that definition, the histories in Our World are defined as maximally modally consistent subsets. These histories represent the possible courses of events in our world—complete space-times—that branch off from one an- other at choice points whose future light cones differentiate the histories. 1090 THOMAS MÜLLER Despite its relativistic soundness and some applications to physical theo- ries ðe.g., Placek 2010Þ, BST has not been much referenced in discussions of space-time theories, even when such discussions are concerned with ques- tions of determinism and indeterminism. In fact, BST92 and some related BST theories have received a fair share of critique on the basis of techni- cal considerations of the space-time models of general relativity ðEarman 2008Þ. Some of that criticism does not apply to BST92 since it only points out conceptual unclarities in relatives of that framework and not in BST92 itself, but there remains an important point of criticism in that BST92 cor- responds poorly to general relativity’s background in differential geometry. While some recent publications ðe.g., Placek and Belnap 2012Þ have begun to address this worry, we agree that so far, no extant formal specification of BST fully meets the critique. This article spells out an overarching framework that is meant to do jus- tice both to the logical aspects of BST and to the physical considerations supporting general relativistic space-time theories. It will be proved that our framework, which deviates from the mentioned theory of BST92 in some technical details, is truly a theory of branching ðin the sense of logicÞ space- times ðin the sense of general relativityÞ. 2. Terminology. When discussing formal models that incorporate modal- ity, such as theories of branching histories, it is important to use terminol- ogy that allows some necessary distinctions. Modal consistency ð“modal flatness”Þ is usually a property of substructures and not of a full structure, which incorporates different alternatives and is therefore meant to be mod- ally inconsistent. We suggest distinguishing • a logical theory, specified via a set of axioms in some formal lan- guage, • models of a logical theory, that is, structures fulfilling the axioms, • a physical theory, which normally should not be identified with some axiomatic framework, • a solution to the equations of the theory, which is a mathematical structure and can often be identified with a history ða complete pos- sible course of eventsÞ, and • the metaphysical notion of a world as something that is unified by “suitable external relations” ðLewis 1986, 208Þ. It is common to identify the notions of model, solution to the equations, history, and ðpossibleÞ world. This identification is not mandatory, how- ever, and in fact positively harmful when it comes to BST. A history ðpos- sible course of eventsÞ indeed has to be “modally flat,” containing no mod- TOPOLOGY FOR BRANCHING SPACE TIMES 1091 ally incompatible events. In the case of BST, models contain more than one history, and each model ðin the logical senseÞ specifies a world ðin the given metaphysical senseÞ. Following Belnap, we propose to call such a world ða model of BSTÞ “Our World” rather than a “possible world,” since the no- tion of a possible world tends to trigger the image of modal consistency ðwhile, to repeat, this is not part of the definition of what a world isÞ. 3. An Overview of BST92. Here we give the axiomatic basis of BST92, due to Belnap ð1992, 2003Þ. Later on we will deviate from this framework in a small but substantial matter of topological detail, in order to move it closer to general relativity. A model of BST92 is a nonempty partial order W; ≤ ih ða nonempty set W together with a transitive, antisymmetric relation ≤ Þ subject to the set of constraints given below.1 Elements of W are called possible point events, or, briefly, events. Let H ⊆ ℘W be the set of maximal upward-directed sub- sets of W. ðIn a partial order, a set is upward directed iff for any two of its elements a and b, there is an element c such that a ≤ c and b ≤ c. We often shorten this to “directed.”Þ Elements of H, that is, maximal directed subsets h ⊆ W , are called histories. A chain in W is a linear subset, that is, a subset c ⊆ W such that for any x; y ∈ c we have either x ≤ y or y < x. The axioms of BST92 are as follows: • W; ≤ ih is a nonempty, dense partial order without maxima. • Each lower-bounded chain C ⊆ W has an infimum in W, written inf C. • Each upper-bounded chain C ⊆ W has a supremum in h ðsuphCÞ for each history h ∈ H for which C ⊆ h. • ðPrior choice principleÞ If C ∈ h 2 h0 is a lower-bounded chain in h none of whose elements is an element of h 0 , then there is a choice point c ∈ h \ h0 such that c is maximal in h \ h0, and c < C ði.e., for all e ∈ C, we have c < eÞ. Note that by the given definition, histories are downward closed: if e ∈ h and f ∈ W such that f ≤ e, then also f ∈ h. Accordingly, if C is a lower- bounded chain in history h, then infC ∈ h as well. As a first link with space-time theories, we can give a generic definition of the causal and the chronological past and future of events in a BST92 model, as follows: Definition 1. Given a BST92 model W; ≤ ih , an event e ∈ W lightlike pre cedes f ∈ W ðin symbols: e ◁ f Þ iff e ≤ f and there is only a single max- 1. We will also use the relation symbol “<,” which is defined in the usual way: x < y if and only if ðiffÞ x ≤ y and x ≠ y. 1092 THOMAS MÜLLER imal chain that has e as its first and f as its last point. Event e chronolog ically precedes f ðin symbols: e ≪ f Þ iff e < f and it is not the case that e ◁ f . On the basis of these notions, we can define the notions of the causal and the chronological future ðand analogously, pastÞ of an event e ∈ W, as usual: Definition 2. Given a BST92 model W; ≤ ih and some e ∈ W, the causal future of e, J1ðeÞ, and the chronological future, I1ðeÞ, are defined as fol- lows: J1ðeÞ:5 f f ∈ W j e ≤ f g; I1ðeÞ:5 f f ∈ W j e ≪ f g: The corresponding past notions are J ðeÞ:5 f f ∈ W j f ≤ eg; I ðeÞ:5 f f ∈ W j f ≪ eg: 4. The Hausdorff Property in Space-Time Theories. A topological space hX; Ri, where X is a nonempty set and the topology R ⊆ ℘X is the col- lection of open sets, is Hausdorff if any two elements of X can be separated by disjoint open sets.2 Hausdorffness forbids, intuitively speaking, the ex- istence of “unseparably close points” or perhaps “doubled points” or “points that occupy the same position.” The branching real line pictured in figure 1 is a simple example of a non-Hausdorff space. Following Hajicek ð1971Þ, who credits Geroch for the notation, we will write xYy to indicate that the points x and y violate the Hausdorff condition, that is, that x and y cannot be separated by disjoint open sets. In figure 1 we have 01Y02. The notation usefully suggests graphically that in such a case, x and y “branch off” from some common trunk, like the left part of the branching lines of figure 1. In fact, in our examples below such x and y will always be different limits of a single converging sequence. As defined, BST92 does not come with a topology. One natural topology has recently been discussed extensively by Placek and Belnap ð2012Þ. With respect to this topology, the Hausdorff property generically fails in models of BST92. Models of BST92 are not locally Euclidean, however, and thus do not form ðgeneralizedÞ manifolds.3 2. For a simple introduction to relevant formal definitions and background, see, e.g., Mendelson ð1990Þ. 3. A manifold is, roughly, a topological space that is locally Euclidean and that, therefore, locally “looks like” Euclidean space of a specific dimension. Malament ð2012Þ gives a nice introduction to this and other notions from differential geometry. Normally, man ifolds are required to be Hausdorff. A generalized manifold is allowed to be non Hausdorff. TOPOLOGY FOR BRANCHING SPACE TIMES 1093 Is non-Hausdorffness a good idea for grounding BST? The discussion of this issue is somewhat tangled. Earman ð2008Þ in his overview of “indi- vidual branching” for single space-times concludes that the only viable path to individually branching space-times comes from non-Hausdorff models because other approaches ðe.g., so-called trousers worldsÞ appear unfeasible. While we follow Earman in this assessment, we want to stress that BST—at least in the form of BST92 and in the form that we are trying to develop fur- ther here—is not meant to give a picture of a single space-time that some- how branches ðperhaps like an amoeba undergoing fissionÞ but to integrate several histories within one logical model. The branching ðoverlappingÞ histories are individually nonbranching and, in fact, Hausdorff space-times. Still, it is useful to look at the physicists’ discussion of non-Hausdorffness that Earman references. In mathematical physics, Hajicek ð1971Þ proves an important result about solutions to the Einstein field equations whose maximal analytic extensions are non-Hausdorff: roughly, a non-Hausdorff space-time either fails to be strongly causal or admits bifurcating geodesics. Hajicek interprets this result as showing that “all such ½i.e., non-Hausdorff � space-times must be weakly acausal” ð75Þ, which would indeed be reason enough for a physicist to shun non-Hausdorff space-times. It is interesting to see how Hajicek supports his interpretation of his theorem. Commenting on bifurcating curves, he writes: “It is easily seen that such curves can only exist in a non-Hausdorff space. Then, if we have some system of ordinary differential equations which has locally a unique solution . . . it is immediate that this system cannot have two different solutions . . . unless these solutions form a bifurcate curve. Therefore, in view of the classical causality conception coinciding with de- terminism it is sensible to rule out the bifurcate curves” ð79Þ. The dialectics is thus as follows: a result from mathematical physics ðHajicek’s theorem 4Þ establishes ðroughlyÞ that in non-Hausdorff space-time models either there is a violation of strong causality or there are bifurcating curves. An appeal Figure 1. Branching real line as a simple non Hausdorff space. A basis for the topology is given by the open intervals in both tracks, reaching into the common part on the left. 1094 THOMAS MÜLLER to determinism rules out the latter; considerations of physicality rule out the former. This amounts to rejecting non-Hausdorff models. We agree with this argument completely. If BST were to give models of a single space-time, these models should not contain bifurcating curves, and most probably they should not be weakly acausal either, so that non- Hausdorffness would be ruled out. However, if one takes up the issue of non-Hausdorff models in order to build formal models for indeterminism, which is what drives the development of BST, then the above argument obviously pulls no weight. The main challenge for bringing together BST and general relativity, in our view, lies not in non-Hausdorffness but in the failure of BST92 to provide generalized manifolds.4 5. Simple Branching: Generalized Minkowskian Manifolds. We move on to the construction of simple branching models that provide a bridge between the logical, order-theoretic point of view of BST ðhistories, i.e., single space-times, as directed setsÞ and the topological point of view of general relativity ða single space-time as a differential manifoldÞ.5 The main technical challenge is to define structures in which two or more space-times are pasted together in such a way that the resulting object is locally Euclid- ean and makes sense as a model of objective indeterminism. In this arti- cle, we discuss the simplest case of such pasting, which is to paste together m Minkowski space-times ðof dimension n > 1Þ at the origin, to arrive at a structure Mnm. There are various choices for this pasting, depending on whether the m origins are represented by one or by m different points in the resulting structure and on how the rim of the future light cone at the origin is handled. BST92’s prior choice principle demands identifying the m or- igins as one point but keeping separate the rim of the future light cones above the origin. This is what breaks local Euclidicity: on such a structure, one cannot define a locally Euclidean topology ðunless one gives up con- nectedness, which is not an interesting optionÞ. Since we want to define generalized manifolds, we have to ascertain local Euclidicity. This means that we need to differ from BST92 in the pasting construction. Our choice is to define structures of m-fold branching, n-dimensional Minkowski space-time Mnm as follows: 6 4. We hereby follow Earman ð2008, 198 99Þ: “topological spaces that are not locally Eu clidean cannot be assigned a differentiable structure, and such a structure is essential in formulating the very notion of a Lorentzian metric and in formulating the Einstein field equations.” Thus, if we want to remain close to general relativity, we had better arrive at a generalized manifold. 5. For reasons of space, the following discussion is quite compressed. For a more de tailed exposition, see Müller ð2011Þ. 6. For a similar construction in the context of BST92, which does not lead to general ized manifolds, see, e.g., Wrónski and Placek ð2009Þ. An Mn m like construction is given TOPOLOGY FOR BRANCHING SPACE TIMES 1095 Definition3 ðMnmÞ. Them-fold branching, n-dimensional Minkowski space- time Mnm is defined from the n-dimensional Minkowski space-time M n ðfor simplicity, we use Rn with the Minkowskian ordering ≤ MÞ by setting a to- be-multiplied region V to be the future light cone of the origin, including the rim of the light cone and the origin itself: V :5 J 1ð0Þ 5 fx ∈ Mn j 0 ≤ Mxg; V :5 M n V; Vi:5 V � fig; defining m layers, for i 5 1; : : : ; m, to be Lni :5 ðV � f1gÞ [ Vi; and pasting them via Mnm :5 ⋃ m i 1 Lni : The ordering ≤ on Mnm is the union of the usual Minkowskian orderings in the layers, and the locally Euclidean topology R is given via the countable basis of open balls with rational center coordinates x and rational radius r in the finitely many layers i 5 1; : : : ; m, Biðx; rÞ :5 fhy; ji ∈ Lni j dðx; yÞ < r & j ∈ f1; igg: Note that the layers themselves, each of which is homeomorphic to Rn, are open sets in this topology and that for i ≠ j, Lni 2 L n j 5 Vi. Note also that yVi 5 yJ1ð0Þ � fig. in Visser ð1996, 251 55Þ; the book contains many pointers to relevant literature. Visser calls his construction a “branched spacetime” ð252Þ, without making any connections to the philosophical/logical discussions about BST, however. Penrose ð1979, 593Þ has a suggestive drawing of a branching space time; while Penrose is not explicit about the topology, and his figure 12.3a may suggest choice points à la BST92, he seems to have our option in mind as well since he writes: “on each branch the wavefunction starts out as a different eigenvector” ð594; italics mineÞ. Deutsch ð1991Þ refers to this discus sion; his remarks about “a larger object which has yet to be given a proper geometrical description” ð3207Þ may be read as pointing in the direction of something like our Mnm structures or their generalizations mentioned below. McCabe ð2005Þ reproduces Pen rose’s figure. He remarks that such figures themselves are open to different interpreta tions and do not need to be read as implying non Hausdorffness; this is in line with our view that there are in fact multiple options for pasting. However, McCabe does not discuss in much detail the price that has to be paid for dropping local Euclidicity in avoiding non Hausdorffness, remarking that “it is a debate which has not been con ducted in the literature” ð670Þ. We agree with Earman that local Euclidicity has to be taken very seriously, and we will continue to hold on to it. 1096 THOMAS MÜLLER 6. Capturing Modal Consistency. Our structures Mnm are still partial or- ders, as in BST92, but we want to move from the order-theoretic to a gen- eralizable, topological characterization of modal consistency. Intuitively and by the pasting construction, the maximal modally consistent subsets of Mnm should be exactly the layers L n i , i 5 1; : : : ; m. These cover the whole of Mn m without any gaps or holes, and they are also individually such that in each layer, each space-time point of Minkowski space-time Mn occurs exactly once. These layers are also the histories in the sense of the usual order-theoretic definition of BST92: each layer is a maximal directed set in Mnm. 7 The question before us is how to capture this intuitive notion of modal consistency in purely topological terms. Hajicek ð1971Þ defines the useful notion of an H-submanifold of a Y- manifold ðwhere the H stands for “Hausdorff,” Y graphically represents non- Hausdorffness as branching, and a Y-manifold is a generalized manifoldÞ: Definition 4 ðH-manifoldÞ. Given a Y-manifold M, a subset A ⊆ M is an H submanifold iff A is open, connected, Hausdorff, and maximal with re- spect to these properties ði.e., every proper superset of A is not open, not connected, or not HausdorffÞ. Hajicek ð1971Þ also suggests to write YL M for the set of points in M that are non-Hausdorff related to some point in L, YL M :5 fx ∈ M j ∃y ∈ L xYyg: On our way toward a useful generalized notion of modal consistency, we note some facts about the points in Mn m that are non-Hausdorff related to some other point ðobviously there are no such points in case m 5 1Þ: Lemma 1. Let M :5 Mnm for some n ∈ N and some m ≥ 2. Then for x 5 hx; ii; y 5 hy; ji ∈ M we have xYy iff x 5 y; i ≠ j; and x ∈ yJ1ð0Þ: Accordingly, YM M 5 fhx; ii j x ∈ yJ1ð0Þ & i ∈ f1; : : :; mgg; and for L :5 Lni a layer ði ∈ f1; : : : ; mgÞ, we have 7. Obviously the layers are directed sets, being order isomorphic to Mn, which is di rected. For maximality, observe that any “new” element to be added to Lni has to come from Vj with j ≠ i; by the definition of the ordering, the resulting superset of Lni is not directed. TOPOLOGY FOR BRANCHING SPACE TIMES 1097 YL M 5 fhx; ji j x ∈ yJ1ð0Þ & j ∈ f1; : : : ; mg & j ≠ ig: Using a further lemma, we can then prove that the layers of Mnm are in fact H-submanifolds:8 Lemma 2. Let M 5 Mnm for some n, m, and let L :5 L n i ⊆ M be a layer ði ∈ f1; : : : ; mgÞ. Then L is an H-submanifold of M. However, the notion of an H-submanifold is not sufficient as an analysis of modal consistency, given that we also want the other direction of lemma 2: Fact 1. M :5 M 22 has an H-submanifold that is not equal to one of the layers Lni , i 5 1; 2. Proof by example. We divide the rim of the forward light cone of the ori- gin into a left and a right part, which are allowed to overlap at the origin: Jl :5 fht; xi ∈ J1ð0Þ j x ≤ 0g; Jr :5 fht; xi ∈ J1ð0Þ j x ≥ 0g: We have Jl [ Jr 5 J1ð0Þ and Jl \ Jr 5 f0g. Now consider the set A :5 M ððJl � f1gÞ [ ðJr � f2gÞÞ; that is, A is the whole of the pasted space M without half of the rim of the forward light cone in each layer. Note that the origin in both layers is removed in constructing A, which makes it intuitively weird. But as a fact, A is an H-submanifold of M. This follows directly from Hajicek ð1971, theorem 2Þ. So, we know that not every H-submanifold can be taken to be a history. By a more detailed consideration of this and other examples, we are finally led to our official topological definition of a maximal consistent set, or a history:9 Definition 5. Given M 5 Mnm for some n and m, a history in M is a subset h ⊆ M that is maximal with respect to the properties of being ðiÞ open, ðiiÞ connected, ðiiiÞ Hausdorff, and ðivÞ for each subset C ⊆ h, if yC ≠ ∅, then h \ yC ≠ ∅ as well. 8. For details, see Müller ð2011Þ. 9. For details, see again Müller ð2011Þ. 1098 THOMAS MÜLLER As a test for the usefulness of this definition, we can now indeed prove both directions of the analogue of lemma 2: Lemma 3. Given M 5 Mnm for some n and m, a subset A ⊆ M is a history according to definition 5 iff A 5 Lni for some i ∈ f1; : : : ; mg. Definition 5 can be applied to any H-manifold; it is not limited to the structures Mnm with respect to which it was motivated. So we have arrived at a general definition of modal consistency in BST. 7. Conclusion. In this article, we have given an overview of the main chal- lenges facing the construction of explicit formal models for indeterminism in a general relativistic setting. The best extant candidate framework for such models is Belnap’s BST92. Some criticisms leveled against the project of con- structing BST seem misplaced, as they rely on the idea that in BST, a single space-time should somehow branch or bifurcate. We agree that this should be avoided. The branching in BST is of a modal nature, and as single space- times are modally consistent, they themselves do not branch. The BST ap- proach, once it is followed with mathematical rigor and once a topology is defined, does lead to non-Hausdorff models, however. Again, a fair share of the criticism of non-Hausdorff models only applies to single space-times and not to BST structures. There is one specific challenge that has not been met so far, however: BST92 does not define generalized manifolds, and there- fore the link with general relativity is not satisfactory. We believe that we have met this challenge, at least for simple structures. In this article, we have defined a method for pasting Minkowski space- times in a locally Euclidean way, such that the resulting structures are gen- eralized manifolds. We have also given a novel definition of modal consis- tency, which is purely topological and which generalizes the order-theoretic definition in terms of directedness on which BST92 is built. A lot remains to be done. So far we have only investigated the simplest structures: pasted Minkowski space-times. We propose that locally, any useful BST should look like one of the Mnm we have defined here. The study of global features of our BST models, however, has to be left for future work. REFERENCES Belnap, N. 1992. “Branching Space-Time.” Synthese 92 ð3Þ: 385 434. . 2003. “No-Common-Cause EPR-Like Funny Business in Branching Space-Times.” Philosophical Studies 114:199 221. Belnap, N., M. Perloff, and M. Xu. 2001. Facing the Future: Agents and Choices in Our Inde- terminist World. Oxford: Oxford University Press. TOPOLOGY FOR BRANCHING SPACE TIMES 1099 Butterfield, J. 2005. “Determinism and Indeterminism.” In Routledge Encyclopedia of Philos- ophy, vol. 3. London: Routledge. Deutsch, D. 1991. “Quantum Mechanics Near Closed Timelike Lines.” Physical Review D 44 ð10Þ: 3197 3217. Earman, J. 2006. “Aspects of Determinism in Modern Physics.” In Handbook of the Philosophy of Physics, ed. J. Butterfield and J. 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Visser, M. 1996. Lorentzian Wormholes. New York: American Institute of Physics. Wroński, L., and T. Placek. 2009. “On Minkowskian Branching Structures.” Studies in History and Philosophy of Modern Physics 40:251 58. 1100 THOMAS MÜLLER Thomas Müllery: space times as a non Hausdorff ðgeneralizedÞ manifold: in Visser ð1996 251 55Þ the book contains many pointers to relevant literature Visser: Text1: Erschienen in: Philosophy of Science ; 80 (2013), 5. - S. 1089-1100 Text2: Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-268027