On the Existence of Spacetime Structure:
Technical Appendices†
Erik Curiel‡

Contents
A Technical Appendix: Limits of Spacetimes

1

B Technical Appendix: Pointless Constructions

4

C Appendix: Observability

8A

Technical Appendix: Limits of Spacetimes

I sketch here the construction of Geroch (1969) (whose exposition I closely follow), which grounds
the arguments of section 3 in Curiel (2016). (I simplify his construction in non-essential ways for our
purposes, and gloss over unnecessary technicalities.) Consider a 1-parameter family of relativistic
spacetimes, by which I mean a family {(Mλ , g ab (λ))}λ∈(0,1] , where each (Mλ , g ab (λ)) is a relativistic
spacetime with signature (+, −, −, −) for g ab (λ). (It will be clear in a moment why I work with thecontravariant form of the metric tensor.) In particular, I do not assume that Mλ is diffeomorphic toMλ0 for λ 6= λ0 . The problem is to find a limit of this family, in some suitable sense, as λ → 0. To
solve the problem in full generality, we will use a geometrical construction, gluing the manifolds Mλof the family together to form a 5-dimensional manifold M, so that each Mλ is itself a 4-dimensional
submanifold of M in such a way that the collection of all of them foliate M.1 λ becomes a scalar† These are technical appendices to the paper “On the Existence of Spacetime Structure” (forthcoming 2016 in

British Journal for Philosophy of Science, early online publication, free access: doi:10.1093/bjps/axw014), working
out details of some of that paper’s constructions and arguments.
‡ textbfAuthor’s address: Munich Center for Mathematical Philosophy, Ludwigstraße 31, Ludwig-MaximiliansUniversität, 80539 München, Germany; email: erik@strangebeautiful.com
1 In general what will result is not a foliation in the strict sense of differential topology, but will rather be a stratified
space (Thom 1969). It is close enough to a foliation, however, to warrant using the more familiar term for simplicity
of exposition, as nothing hinges on the technical differences between the two.

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Technical Appendicesfield on M, and the metrics g ab (λ) on each submanifold fit together to form a tensor field g AB on M,
of signature (0, +, −, −, −). (I use majuscule indices for objects on M.) The gradient of λ on M
determines the singular part of g AB : g AN ∇N λ = 0. (This is why I work with the contravariant formof the metric; otherwise, we could not contravect its five-dimensional parent in any natural way withthe gradient of λ.) Note that g AB by itself already determines the submanifolds Mλ (viz., as thesurfaces defined by g AN ∇N λ = 0), and that it does so in a way that does not fix any identificationof points among them. In other words, the structure I posit does not allow one to say that a point in
Mλ is “the same point in spacetime” as a point in a different Mλ0 (as I shall discuss at some length
below).
To define a limit of the family now reduces to the problem of the attachment of a suitable boundary to M “at λ = 0”. A limiting envelopment for M, then, is an ordered quadruplet (M̂, ĝ AB , λ̂, Ψ),
where M̂ is a 5-dimensional manifold with paracompact, Hausdorff, connected and non-trivial boundary ∂ M̂, ĝ AB a tensor field on M̂, λ̂ a scalar field on M̂ taking values in [0, 1], and Ψ a diffeomorphismof M to the interior of M̂, all such that
1. Ψ takes g AB to ĝ AB (i.e., Ψ is an isometry) and takes λ to λ̂
2. ∂ M̂ is the region defined by λ̂ = 0
3. ĝ AB has signature (0, +, −, −, −) on ∂ M̂This makes precise the sense in which M̂ represents M with a boundary attached in such a waythat the metric on the boundary (ĝ AB restricted to ∂ M̂) can be naturally identified as a limit of themetrics on the Mλ (g AB on M). I call {(Mλ , g ab (λ))}λ∈(0,1] an ancestral family of the spacetime
represented by ∂ M̂, and I call ∂ M̂ the limit space of the family with respect to the given envelopment.
In general, a given spacetime will have many ancestral families, and an ancestral family will have
many different limit spaces. For the sake of convenience I will often not distinguish between M andthe interior of M̂. (Although it is tempting also to abbreviate ‘∂ M̂’ by ‘M0 ’, I will not do so, becausepart of the point of the construction is that different spacetimes can have the same ancestral family.)
To characterize the metrical structure of the limit space using structure of members of theancestral family, I introduce one more construction. An orthonormal tetrad ξ(λ) at a point pλ ∈ Mλis a collection of 4 tangent vectors at the point mutually orthogonal with respect to gab (λ). Let γ
be a smooth curve on M nowhere tangent to any Mλ such that it intersects each exactly once. γ
then is composed of a set of points pλ ∈ Mλ , one for each λ. A family of frames along γ is a familyof orthonormal tetrads, one at each point of the curve such that each vector in the tetrad is tangent
to its associated Mλ , whose members vary smoothly along it. In general, a family of frames will
have no well defined limit in M̂ as λ → 0, i.e., there will be no tetrad ξ(0) at a point of ∂ M̂ that thefamily ξ(λ) converges to; in this case, I say the family is degenerate. It is always possible, however,given a tetrad ξ(0) at a point on the boundary to find some family of frames that does converge toit.
Now, fix ξ(0) at p0 ∈ ∂ M̂ and a family of frames ξ(λ) that converges to it. We can represent
the metric tensor gab (λ) in a normal neighborhood of pλ in Mλ using the normal coordinate system
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Technical Appendicesthat ξ(λ) defines in the neighborhood. In a normal neighborhood of p0 , the components of themetric with respect to these coordinates converge as λ → 0, and the limiting numbers are just thecomponents of gab (0) at p0 with respect to the normal coordinates that ξ(0) defines. In this way,
we can characterize all structure on the limit space based on the behavior of the corresponding
structures along the family of frames in the ancestral family.
I turn now to an example immediately relevant to my arguments.

Consider a family{(Mλ , g ab (λ))} of Reissner-Nordström spacetimes each element of the family having the same fixedvalue M for its mass and all parametrized by electric charge λ, which converge smoothly to 0.2
Construct their envelopment. One can now impose a natural collection of families of frames on
the family, with Schwarzschild spacetime as the limit.3 Now, comparison of figures 3.1 and 3.2 inCuriel (2016) suggests that something drastic happens in the limit. All the points in the throat of
the Reissner-Nordström spacetimes (the shaded region in the diagram) seem to get swallowed by
the central singularity in Schwarzschild spacetime—in some way or other, they vanish. Using ourmachinery we can make precise the question of their behavior in the limit λ → 0 in the envelopment.
Consider the points in the shaded region in figure 3.2 in Curiel (2016), between the lines r = 0
and r = r− . (r is the radial coordinate in a system that respects the spacetime’s spherical symmetry;the coordinate values r− and r+ define boundaries of physical significance in the spacetime, whichin large part serve to characterize the central region of the spacetime as a black hole.) Fix a natural
family of frames along a curve in M composed of points qλ each of which lies in the shaded region in
its respective spacetime. It is straightforward to verify that the family of frames along the curve does
not have a well defined limit: roughly speaking, the curve runs into the Schwarzschild singularity
at r = 0. In this sense, no point in Reissner-Nordström spacetime to the future of the horizonr = r− has a corresponding point in the limit space. To sum up: one begins with a family of
Reissner-Nordström spacetimes continuously parametrized by electric charge, which converges to 0,
and constructs the envelopment of the family; one constructs the limit space by a choice of familiesof frames; the collection of families of frames enforces an identification of points among different
members of the family of spacetimes, including a division of those points that have a limit fromthose that do not; and that identification, in turn, dictates the identification of spacetime points
in the limit space (which points in the ancestral family lie within the Schwarzschild radius, e.g.,
and which do not). Thus one can identify points within the limit Schwarzschild spacetime, one’s
idealized model, only by reference to the metrical structure of members of the ancestral family; one
can, moreover, identify points in the limit space with points in the more complex, initial models
one is idealizing only by reference to the metrical structure of the members of the ancestral family
as well. It is only by the latter identification, however, that one can construe the limit space as
an idealized model of one’s initial models, for the whole point is to simplify the reckoning of the
2 I ignore the fact that electric charge is a discrete quantity in the real world, an appropriate idealization in thiscontext.
3 The frames are natural in the sense that they conform to and respect the spherical and the timelike symmetries
in all the spacetimes. One could use this fact to explicate the claim that Schwarzschild spacetime is the canonicallimit of Reissner-Nordström spacetime, in the sense that it is what one expects on physical grounds, whatever exactlythat may come to, in the limit of vanishing charge while leaving all else about the spacetime fixed.

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Technical Appendices

physical behavior of systems at particular points of spatiotemporal regions of one’s initial models.
One can, moreover, use different families of natural frames to construct Schwarzschild spacetime
from the same ancestral family, with the result that in each case the same point of Schwarzschildspacetime is identified with a different family of points in the ancestral family. More generally,
different families of frames will yield limit spaces different from Schwarzschild spacetime, with nocanonical way to identify a point in one limit space (one idealized model the theoretician constructs)
with one in another. In other words, the identification of points in the limit space depends sensitively
on the way the limit is taken, i.e., on the way the model is constructed. In consequence, in so
far as one conceives of Schwarzschild spacetime as an idealized model of a richer, more complete
representation, one can identify points in it only by reference to the metrical structure of one of its
ancestral families, and one can do that in a variety of ways.
Every spacetime has at least one ancestral family, the trivial one consisting of the continuous
sequence of itself, so to speak. Construct an envelopment M for it, with it itself as the limit space,and apply a slight twist, so to speak, to every metric in every model in the family so as to rendereach model non-isometric to any other, i.e., so as to render the family non-trivial. (One can makethis idea precise in any of a number of simple ways, such as using a smoothly varying 1-parameterfamily of homotopies or linear perturbations.) On a curve in M, fix a family of frames that has awell defined limit on ∂ M̂. Now, define a family of Lorentz transformations along that curve, one
transformation at each point, such that the family varies smoothly along the curve, and such thatwhen one applies each transformation to the tetrad at its point, the result is a family of framesthat has no well defined limit. (One can always do this; for example, the Lorentz transformations
can cause the tetrads to oscillate wildly as λ → 0.) The points of the ancestral family along thatcurve have no corresponding point in the limit space defined by the resulting family of frames. This
proves:
Proposition A.1 Every spacetime has a non-trivial ancestral family with vanishing points. Every
non-trivial ancestral family has a limit space with respect to which some of its points vanish.

B

Technical Appendix: Pointless Constructions

I give here the constructions relevant to the arguments of section 4 in Curiel (2016). The basic ideaof the construction of a pointless manifold is simple. I posit a class of sets of rational numbers to represent the possible values of physical fields, with a bit of additional structure in the form of primitive
relations among them just strong enough to ground the definition of a derived relation whose naturalinterpretation is “lives at the same point of spacetime as”. A point of spacetime, then, consists of anequivalence class of the derived relation. The derived relation, moreover, provides just enough rope
to allow for the definition of a topology and a differential structure on the family of all equivalenceclasses, and from this the definition of all tensor bundles over the resultant manifold, completingthe construction. The posited primitive and derived relations have a straightforward physical interpretation, as the designators of instances of a schematic representation of a fundamental type of

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Technical Appendicesprocedure the experimental physicist performs on physical fields when he attempts to ascertain relations of physical proximity and superposition among their observed values. An important example
of such an experimental procedure is his use of the observed values of physical quantities associatedwith experimental apparatus to determine the values of quantities associated with other systems,
those he investigates by use of the apparatus. This interpretation of the relations motivates theclaim that the constructed structure suffices, for our purposes, as a representation of spacetime inthe context of a particular type of experimental investigation as modeled by mathematical physics,
and is not (only) an abstract mathematical toy. Because of limitations of space, I give only a baresketch of the construction. SeeCuriel (2015)
I begin the construction by laying down some definitions. Let Q be the set of rational numbers.
]
A simple pointless field φ (or just simple field ) is a disjoint union
fp , indexed by the set Q4 ,p∈Q4such that
1. every fp ∈ Q
2. there is exactly one fp ∈ φ for each p ∈ Q4
3. there are two strictly positive numbers Bl and Bu such that Bl < |fp | < Bu for all p ∈ Q4
4. the function φ̄ : Q4 → Q defined by φ̄(p) = fp is continuous in the natural topologies on thosespaces, except perhaps across a finite number of compact three-dimensional boundaries in Q4
Our eventual interpretation of such a thing as a candidate result for an experimentalist’s determination of the values for a physical field motivates the set of conditions. That we index φ overQ4 means we assume that the experimentalist by the use of actual measurements and observations
alone can impose on spacetime at most the structure of a countable lattice indexed by quadruplets ofrational numbers (and even this only in a highly idealized sense); in other words, the spatiotemporal
precision of measurements is limited. Condition 1 says that all measurements have only a finite precision in the determination of the field’s value. Condition 2 says that the field the experimentalist
measures has a definite value at every point of spacetime. Condition 3 says that there is an upper
and a lower limit to the magnitude of values the experimentalist can attribute to the field using
the proposed experimental apparatus and technique; for instance, any device for the measurementof the energy of a system has only a finite precision, and thus can attribute only absolute values
greater than a certain magnitude, and the device will be unable to cope with energies above a given
magnitude. Condition 4 tries to capture the ideas that (local) experiments involve only a finite
number of bounded physical systems (apparatuses and objects of study), and that classical physicalsystems bear physical quantities the magnitudes of which vary continously (if not more smoothly),
except perhaps across the boundaries of the systems.
Fix a family Φ of simple pointless fields. The link at p, λp , is the set containing the elementsfrom all simple fields in Φ indexed by p ∈ Q4 . A linked family of simple pointless fields F is anordered pair (Φ, Λ) where Φ is a collection of distinct simple fields, and Λ is the family of links on
Φ, a linkage, complete in the sense that it contains exactly one link for each p ∈ Q4 . The idea is that
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Technical Appendicesthe values of the simple fields in the same link all live “at the same point of spacetime”, namely thatdesignated by p. One can think of the linkage as a coordinate system on an underlying, abstract
point set.
We are almost ready to define the point-structure of the spacetime manifold. We require only
a few more definitions and two more constructions, which I give in an abbreviated fashion so as toconvey the main points without getting bogged down in unnecessary technical detail. First, given
an open set O ∈ Q4 and a simple pointless field φ, let φO be the field restricted to fp for p ∈ O.
Given a family of simple pointless fields Φ, denote by ‘ΦO ’ the family of simple pointless fields in Φ
restricted to O, and similarly for ‘ΛO ’, defined in the obvious way. Now, let F = (Φ, Λ) be a linked
family containing a countable number of simple fields; we call it a simple fundamental family. Let
F̂ = (Φ̂, Λ̂) be another. Let FO be the family of simple pointless fields and the linkage in F restricted
to the open set O. We want a way to relate the linkages of FO and F̂Ô , for open sets O and Ô,so as to be able to represent the relation between the coordinate systems of two different charts on
the same neighborhood of the spacetime manifold, or on the intersection of two neighborhoods. A
cross-linkage between two simple fundamental families F and F̂ is an ordered triplet (O, Ô, χ) where
O, Ô ⊆ Q4 are open sets, such that either both are the null set or else both are homeomorphic to
Q4 , and χ is a homeomorphism of O to Ô. The link λp ∈ Λ for p ∈ O, then, will designate the same
point in the underlying manifold as λ̂χ(p) ∈ Λ̂ for χ(p) ∈ Ô; in this case, we say the links touch. If
O and Ô are the null set, then the represented neighborhoods do not intersect. (We do not requirethat the values of the scalar fields in the two different simple fundamental families be numericallyequal at any given point, as the two scalar fields may represent different physical quantities, e.g.,
a component of the fluid velocity and a component of the shear-stress tensor of a viscous fluid.) F
and F̂ are to represent coordinate charts on open sets of the underlying spacetime manifold, and thecross-linkage the relation between the ways that FO and F̂Ô respectively “assign coordinates” to thesame spacetime region, viz., the one defined by the intersection of the “domains” of F and F̂. (Theidea of a cross-linkage can be extended to cover more than two simple fundamental families in theobvious way.)
To finish the preparatory work, we must move from rationals to reals. Fix a simple fundamental
family F containing all simple pointless fields, a complete simple fundamental family. First, we
attribute to F the algebraic structure of a module over Q. For example, the sum of two simple
pointless fields φ and ψ in Φ is a simple pointless field ξ such that xp ≡ fp + gp is the value in ξlabeled by the index p, where fp ∈ φ and gp ∈ ψ. ξ is clearly itself a simple pointless field with anatural embedding in the linkage on F, and so belongs to Φ. Now, roughly speaking, we take a double
Cauchy-like completion of Φ over both the points p ∈ Q4 and the values fp̂ ∈ Q, yielding the familyΦ̄ of all disjoint unions of real numbers continuously indexed by quadruplets of real numbers.4 This
procedure makes sense, because every continuous real scalar field on R4 is, again roughly speaking,the double limit of some sequence of bounded, continuous rational fields defined on Q4 . We thus
4 In order to get the completion we require, standard Cauchy convergence does not in fact suffice. We must rather

use a more general method, such as Moore-Smith convergence based on topological nets. The technical details are
not important. See, e.g., Kelley (1955, ch. 2) for details.

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Technical Appendicesobtain what is in effect the family Φ̄ of all continuous real scalar fields on R4 , though I refer to themas pointless fields, in so far as, at this point, they are still only indexed disjoint unions. The limiting
procedure, moreover, induces on Φ̄ the structure of a module over R from that on Φ. Finally, inthe obvious way, we take the completion, as it were, of Λ using the same limiting procedure toobtain a linkage Λ̄ on Φ̄. I call F̄ = (Φ̄, Λ̄) a complete fundamental family. A cross-linkage on a pairof fundamental families is the same as for simple fundamental families, except only that one uses
homeomorphisms on subsets of R rather than Q. If we have two simple fundamental families witha cross-linkage on them and take limits to yield two fundamental families, then the nature of thelimiting process guarantees a unique cross-linkage on the two fundamental families consistent withthe original.
We can at last construct a real topological manifold from a collection a cross-linkage on a familyof simple fundamental fields. The basic idea is that a complete fundamental family represents thefamily of continuous real functions on the interior of a bounded, normal neighborhood of what
will be the spacetime manifold. Because a spacetime manifold must be paracompact (otherwise it
could not bear a Lorentz metric), there is always a countable collection of such bounded, normalneighborhoods that cover it. This suggests
Definition B.1 A pointless topological manifold is an ordered pair ({Fi }i∈N , χ) consisting of acountable set of complete simple fundamental families and a cross-linkage on them.
To justify the definition, I sketch the construction of the full point-manifold and its topology. First,
we take the joint limit of all simple fundamental families to yield a countable collection of fundamental families with the induced cross-linkage. A point in the manifold, then, is an equivalence class oflinks, at most one link from each family, under the equivalence relation “touches”. The set of links
associated with one of the families, then, becomes a representation, with respect to the equivalence
relation, of the interior of a compact, normal neighborhood in the manifold, and the fields in thatfamily represent the collection of continous real functions on that neighborhood. The cross-linkage
defines the intersections among all these neighborhoods, yielding the entire point-set of the manifold. By assumption, the collection of all such neighborhoods forms a sub-base for the topologyof the manifold, and so, by constructing the unique topological base from the given sub-base, thepoint-set becomes a true topological manifold. It is straightforward to verify, for example, that areal scalar field on the constructed manifold is continuous if and only if its restriction to any of thebasic neighborhoods defines a field in the fundamental family associated with that neighborhood.
Now, to complete the construction, we can define the manifold’s differential structure in astraightforward way using similar techniques. First, demarcate the family of smooth scalar fields asa sub-set of the continuous ones, which one can do in any of a number straightforward ways withclear physical content based on the idea of directional derivatives. (The algebraic modular structureof the fields comes into play in the definition of the directional derivative.) The family of all smoothscalar fields on a topological manifold, however, fixes its differential structure (Chevalley 1947). The
directional derivatives themselves suffice for the definition of the tangent bundle over the manifold,
and from that one obtains all tensor bundles.
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Technical AppendicesC

Appendix: Observability

This discussion supplements that of section 6 in Curiel (2016).
One does not have to be an instrumentalist or an empiricist to accept that the possible observability of physical phenomena is one of the most fundamental reasons we have to think such things
are physical in the first place. The question of the observability of various kinds of global structure in
general relativity, therefore, poses particularly interesting problems for arguments about physicality.
Manchak (2009, 2011) shows that, in a precise sense, local observations can never suffice to determine
the complete global structure of spacetime in general, and in particular cannot determine whether aspacetime is inextendible or stably causal (Manchak 2011, p. 418, proposition 3). Nonetheless, there
remain several things to say and ask about the matter of physicality here.
Take, for example, the Euler number of the spacetime manifold, a global topological structure.5
It is a topological invariant that, in part, constrains the possible existence of everywhere non-zero
vector fields on a manifold. That an even-dimensional sphere, for example, possesses no everywhere
non-zero vector field (and indeed no Lorentzian metric) follows directly from the computation of
its Euler number. If we were to live in a world whose underlying manifold possessed a non-trivial
Euler number, and so could support no physical process that would manifest itself as an everywhere
non-zero vector field, this would constitute a physical fact about the world in an indubitable sense.
It is not clear to me, however, whether in some precise sense the Euler number of the spacetime
manifold could ever be determined by direct observation.
The orientability of spacetime is an example of a global topological structure that seems to bestrictly inobservable in an intuitive sense. This follows from the fact that one can construct anorientable manifold from any non-orientable one by lifting the structures on it to a suitable covering
space, which is automatically orientable. The lift of the spacetime metric to a covering manifold,however, would yield a representation of exactly the same physical spacetime as the original: every
physical phenomena in the one has an isometric analogue, as it were, in the other, and vice-versa.
Whether or not a spacetime manifold is simply connected, moreover, seems to be in the same boat,
for the universal covering manifold of a manifold is guaranteed to be simply connected.6
Nonetheless, I think those answers about the possible observability of a manifold’s orientability
and simple connectedness may be too pat. If one were to observe that any member of a certain
5 See, e.g., Alexandrov (1957, ch. viii).
6 In

order for a manifold to possess a universal covering manifold, it must be semi-locally simply connected.
Intuitively, this means that it cannot contain “arbitrarily small holes”. More precisely, it means that every pointin the space has a neighborhood such that every loop in the neighborhood can be continuously contracted to apoint. (The contraction need not occur entirely with the given neighborhood.) The so-called Hawaiian Ear-Ring is
an example of a topological space that is not semi-locally simply connected (Biss 2000). Whether or not a spacetime
manifold is semi-locally simply connected presents us with yet another type of question related to physicality: strictly
speaking, there is no physical need for a manifold to possess a universal cover, and it is difficult, to say the least,
to see what other physical relevance being semi-locally simply connected could have; and yet the construction of theuniversal cover is such an extraordinarily useful theoretical device (Geroch 1967) that one wants to demand that acandidate spacetime manifold be semi-locally simply connected. What status does such a demand have? A purely
pragmatic one?

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Technical Appendicesfamily of closed, physically distinguished spatiotemporal loops could not be continuously deformed
into any member of another family of closed, physically distinguished spatiotemporal loops, one
would have shown that the spacetime manifold is not simply connected. Similarly, if one could
show that to parallely propagate a fixed orthonormal tetrad around a given closed spatiotemporalloop would result in its inversion, one would have demonstrated that spacetime is not orientable. I
personally have no idea what sorts of experiment could show either of those things. The history ofphysics, however, if it shows us nothing else, does show us never to underestimate the ingenuity ofexperimentalists, no matter what the theoretician may tell them is impossible to observe or measure.
The first Betti number of the spacetime manifold offers another interesting example of this sort.
The first Betti number of a topological space is the number of distinct connected components it
has; any manifold with a first Betti number greater than one is ipso facto not connected. Say that
we posited a non-connected spacetime manifold. According to the principles of general relativity,
any phenomena in one component would be strictly inobservable in any other. By this criterion, it
makes no sense to attribute physicality to regions of spacetime disconnected from our own.
So, are these possibly inobservable global structures physical? Well, it seems to me that in somesenses they are, and in others they are not. The only lesson I want to draw here is that questionsof this sort require in-depth investigation sensitive both to the technical details of the mathematics
and to the physical details of how such structures may and may not bear on other phenomena wethink of as manifestly physical, even if they turn out to be indubitably inobservable.7

References
Alexandrov, P. (1957). Combinatorial Topology: The Betti Groups, Volume 2. Rochester, NY:
Graylock Press. A translation by H. Komm of the first Russian edition of 1947.
Biss, D. (2000). A generalized approach to the fundamental group. American Mathematical
Monthly 107 (8), 711–720. doi:10.2307/2695468.
Chevalley, C. (1947). Theory of Lie Groups i. Princeton, NJ: Princeton University Press. 8thprinting, 1970.
Curiel, E. (1999). The analysis of singular spacetimes. Philosophy of Science 66, S119–S145.
Supplement. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers. Stable URL: http://www.jstor.org/stable/188766. A
more recent, corrected, revised and extended version of the published paper is available at:
http://strangebeautiful.com/phil-phys.html.
Curiel, E. (2015). On the existence of spacetime structure: Technical appendices. This paper consists of technical appendices to the paper “On the Existence of Spacetime Structure” (forthcoming 2016 in British Journal for Philosophy of Science), working out details of some of
7 The family of phenomena in relativistic spacetimes grouped under the rubric “singular stucture” (or “singulari-ties”) provides on its own a rich and diverse selection of examples, which I do not have room even to sketch here. See
Curiel (1999) for an extended discussion.

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Technical Appendicesthat paper’s constructions and arguments. URL: http://strangebeautiful.com/papers/
curiel-exist-st-struct-tech-apdx.pdf.
Curiel, E. (2016). On the existence of spacetime structure. British Journal for the Philosophy of Science. doi:10.1093/bjps/axw014. Preprint: arXiv:1503.03413 [physics.hist-ph]. A
manuscript containing technical appendices working out details of some of the constructions and arguments, and containing further discussion of the possible observability of different kinds of spacetime structure, is available at http://strangebeautiful.com/papers/
curiel-exist-st-struct-tech-apdx.pdf.
Geroch, R. (1967). Topology in general relativity. Journal of Mathematical Physics 8 (4), 782–786.
doi:10.1063/1.1705276.
Geroch, R. (1969). Limits of spacetimes. Communications in Mathematical Physics 13 (3),
180–193. doi:10.1007/BF01645486. Open access at http://projecteuclid.org/euclid.cmp/
1103841574.
Kelley, J. (1955). General Topology. The University Series in Higher Mathematics. Princeton:
D. Van Nostrand Company, Inc.
Manchak, J. (2009, January). Can we know the global structure of spacetime? Studies in History
and Philosophy of Modern Physics 40 (1), 53–56. doi:10.1016/j.shpsb.2008.07.004.
Manchak, J. (2011, July). What is a physically reasonable spacetime? Philosophy of Science 78 (3),
410–420. doi:10.1086/660301.
Thom, R. (1969). Ensembles et morphismes stratifiés. Bulletin of the American MathematicalSociety 75, 240–284. doi:10.1090/S0002-9904-1969-12138-5.

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