8.2 The classification of curves via their limit points
8.2.1 Introductory material
Before we begin we note that limit points, as defined in definition 3.1.10, are a subset of the topological limit points, A.1.2, of the curve. There exist cases where the curve converges to itself and therefore it is possible for points on the curve to also be limit points. As a consequence the set of limit points can, in some cases, be the same as the set of topological limit points. The curveγ of subsection 8.3.1 provides an example. Since we shall work so closely with limit points, in this chapter, it is important to keep these subtleties in mind.
8.2.1 Definition. Given γ a curve in a topological space X, let Wγ or simply W
be the set of limit points of γ.
Using this definition we have the following result which illustrates howW relates to topological closure.
8.2.2 Lemma. Let γ : [0, a) → X be a curve in a topological space X. Then
γ =γ∪W.
Proof. It is clear that γ∪W ⊂γ. Let x∈ γ. If x∈γ then x ∈γ∪W, so suppose that x ∈ γ\γ then there exists {si = γ(ti)} a sequence in γ so that {si} → x. If
{si}is full then x∈W, so suppose that{si} is not full. Then there existst ∈[0, a)
so that ti →t. Sinceγ is continuous, then {γ(ti)} →γ(t), henceγ(t) =x, and thus
x∈γ, a contradiction. Therefore γ ⊂γ∪W.
8.2.3 Definition. We say that a curveγ in a topological spaceXis a winding curve if Wγ has at least two distinct elements.
8.2.2
The classification
For the rest of this section we shall assume thatMis a manifold, andγ : [0, a)→ M is a curve. We shall classify γ via the cardinality of W.
8.2 The classification of curves via their limit points 125
Proof. Certainly it is possible for |W| to equal 0 or 1. So suppose that p, q ∈ W
and thatp6=q. Let N be a compact neighbourhood ofp, so thatpis in the interior of N, int (N), and so that q 6∈ N. Let x = {xi = γ(txi)} and y = {yi = γ(tyi)}
be full sequences in γ so that x → p, y → q, x ⊂ N and tx i < t
y
i. For each i, let
γi : [0, bi] → M be the arc length parametrised segment of γ from xi to yi that
is contained in N. That is γi = γ|[tx i,t
y
i]∩N. Then by lemma 8.1.1 there exist a series ofC∞ functionsf
i :X →Yi so that there is a subsequence,{γkifki}, of{γifi} that uniformly converges, on compact subsets of X, to a curve λ :X → Mso that
λ⊂N.
As λ is continuous, if there exists u ∈ X so that λ(0) 6= λ(u) then |λ(X)| = |R|. Since λ(X) ⊂ W ⊂ M we know that |R| = |λ(X)| ≤ |W| ≤ |M| = |R|, thus
|W|=|R|. So suppose that for all u∈X, λ(u) =λ(0).
For each i, by construction, we know that γi ∩(N −int (N)) is non-empty. Let
tvi ∈ X so that γifi(tvi) ∈ N −int (N). As N is compact, the sequence {γi(tvi)}
must have a limit point v. By construction v ∈N −int (N), and asp∈int (N) we know that v 6=p. For each tv
i, we have the sequence {xij =γkjfkj(t
v
i)}j which must
converge to xi =λ(tv
i) as tvi is contained in a compact subset of X. Choosing d to
be a complete distance on Mwe know that
∀ >0∃n()>0 :i > n⇒d(γifi(tvi), v)< ∀ >0∃m()>0 :j > m⇒d(γkjfkj(t v i), λ(t v i))< , and as {γkjfkj(t v i)} →λ(tvi) we know that ∀ >0∃k()>0 :p, q > k ⇒d(γkqfkq(t v i), γkpfkp(t v i))< .
Let >0 choosep >max{n 3, k 3}and q >max{m 3, k 3}. Then we have the following calculation
d(λ(tvkp), v)≤d(λ(tvkp), γkq(t v kp)) +d(γkq(t v kp), γkp(t v kp)) +d(γkp(t v kp), v) = 3+ 3+ 3 =. That is {λ(tv
kp)} →v. By assumption, however, λ(X) ={λ(0)}and therefore asM is hausdorff λ(0) = v. This is a contradiction, therefore there exists u∈ X so that
λ(u)6=λ(0) and we have our result.
We note that for this proof to work it is only necessary that the sequence ydoes not havep as a limit point. Then the compact neighbourhoodN will exist and the rest of the proof follows as written. In particular, if yhas no cauchy subsequences, then such an N will exist. That is if we have x→p and y with no cauchy subsequences then |W|=|R|.
Using this result, we are now able to classify curves in space-time into three types;
|W|= 0,1 or |R|.
8.2.3
|W|
= 0
These curves are extremely well behaved and understood. Any inextendible geodesic in Minkowski space-time is of this type.
8.2.5 Definition. A setU in a topological space is precompact if U is compact. 8.2.6 Definition. A curve γ : [0, a) → T in a topological space T is partially imprisoned if there exists a compact set K ⊂ T and a full sequence, s in γ so that s⊂K.
8.2.7 Definition. A curve γ : [0, a) → T in a topological space T is eventually imprisoned if there exists a compact set K ⊂ M and b ∈[0, a) so that γ|[b,a) ⊂K. Note that if γ is eventually imprisoned then it must also be partially imprisoned. 8.2.8 Proposition. If |Wγ| = 0 then γ is inextendible, not precompact and not
partially nor eventually imprisoned.
Proof. If γ was extendible then there would exist λ : [0, b) → M so that a ∈ [0, b) and for all 0 ≤ t < a, λ(t) =γ(t). This implies, however, that W ={λ(a)}, which is a contradiction. Let {xi} be a full sequence in γ, then since |W| = 0, {xi} has
no convergent subsequence. Therefore γ is not precompact. Suppose that γ was partially imprisoned or eventually imprisoned. By definition (see 8.2.6 or 8.2.7) there must exist x∈W, but this is a contradiction.
8.2 The classification of curves via their limit points 127
8.2.4
|W|
= 1
We may think of these curves as those that can be extended. 8.2.9 Proposition. If Wγ has one element then γ is precompact.
Proof. Suppose that W ={p} has one element and suppose that γ is not precom- pact. Then there exists a full sequence x in γ so that x has no limit points. This being the case, there must also exist a full sequence yin γ so that y→p. Let d be a complete distance on M and choose U ∈ N(p) so that U is compact. If x ⊂ U
then x must have some limit point, as U is compact. Since this is a contradiction, we know that x∩U is finite. Thus we can choose V ∈ N(p) so that x∩V =∅. In this situation we can employ the techniques of 8.2.4 to show that |W|=R. This is a contradiction and therefore γ is precompact.
8.2.10 Corollary. Suppose that γ is of type |Wγ| = 1. Then γ is eventually im-
prisoned.
Proof. From proposition 8.2.9, γ is compact. Thus for any compact set K so that
γ ⊂K, we know that γ is eventually imprisoned in K.
8.2.11 Corollary. If Wγ ={p} then p is the endpoint of γ : [0, a)→ M. That is
every full sequence in γ as p as a limit point.
Proof. If there exists a full sequence with limit pointq6=por with no limit point then
|Wγ|=|R| as shown in the proof of proposition 8.2.9. Since this is a contradiction
then γ must have p as an endpoint.
8.2.12 Proposition. The curveγ : [0, a)→ Mis extendible if and only ifWγ ={p}
and limt→aγ0 exists and is non-zero.
Proof. Suppose that γ : [0, a) → M is extendible and let µ : [0, b) → M be a curve so that µ|[0,a) = γ. By definition 3.1.8, µ0 is non-zero for all t ∈ [0, b), thus limt→aγ0 =µ0(a) = 0. It is also clear that Wγ ={µ(a)}.
Conversely suppose that Wγ = {p} and limt→aγ0 exists and is non-zero. From
corollary 8.2.11 we know that γ(t)→ p. Let v = limt→aγ0 ∈ TpM, then v 6= 0 and
we can choose a curve λ : [0, b) → M, so that λ0(0) = v. We can extend γ, by adjoining λ toγ, re-parameterising and smoothing if necessary.
Note that up until proposition 8.2.12 we have only used results needing our curves to be C0. Proposition 8.2.12, however, has the form it does because of definition 3.1.8. That is our curves must be C1 with non-zero tangent vector, hence the requirement that the limit tangent vector exist and be non-zero in proposition 8.2.12. If we relax definition 3.1.8, to allow for C0 piecewise C1 curves, the condition that limt→aγ0 exists and is non-zero can be replaced with the condition that there exists
a reparametrisation, s of γ so that γ◦s, rather than γ, can be extended.
We note that various differentiability conditions on γ can be satisfied for the curve
µ given by adjoining λ to γ. This can be done because, other than λ0(0) = v we have no conditions on µ. In particular, ifγ is Cn then λ can be chosen so that µis
Cn.
8.2.5
|W|
=
|R|
These curves can prove problematic when working with the Abstract Boundary. Chapter 9 provides a specific example by showing how their presence prevents a simple extension of causality to the Abstract Boundary. Intuitively, these curves revisit a point infinitely many times.
We give a number of examples of these curves in the next section.
8.2.13 Proposition. Let γ be a curve in M. If γ is of type |W| = |R| then γ is inextendible.
Proof. Suppose that |W| = |R| then, from proposition 8.2.12, we know that γ is inextendible.
We note that when using C0, piecewise C1, curves a similar result to proposition 8.2.13 can be proven using the altered form of proposition 8.2.12 mentioned above. As we shall see in subsection 8.3.3, there are both precompact and non-precompact curves of this type. Note that for boundaries such as the g, c, and b-boundary it is the precompact curves of this class that create non-Hausdorff behaviour; see the excellent review in [3] for details.
8.2.14 Proposition. Let γ : [0, a) → M be a curve and let d : M × M → R be a complete distance on M. If the curve γ is of type |Wγ| = |R|, then there exists
an open set U, with U compact, so that γ is partially imprisoned, but not eventually imprisoned, in U.
8.2 The classification of curves via their limit points 129
Inextendible
Not precompact
Extendible Precompact Eventually
imprisoned Partially imprisoned Not imprisoned Either precompact or not precompact |W|=|R| |W|= 0 |W|= 1
Figure 8.1: Summary of the classification ofγ. Read the figure from left to right, so that curves of type |W|= 1 are extendible, precompact and eventually imprisoned. Note that we have left off a few technical details, such as limt→aγ0 exists and is non-zero.
Proof. Suppose that |Wγ| = |R|, and let {si = γ(tsi)} be a full sequence in γ so
that s converges to x ∈ W. Choose U ∈ N(x) so that U is compact and there exists y ∈ W −U. We can choose a full sequence {pi = γ(tpi)} in γ so that for
all i, ts i < t
p
i < tsi+1 and {pi} → y. Then there exists i∗ so that for all j > i∗,
γ(ts
j), γ(tsj+1)∈ U and γ(t
p
j) 6∈ U, with tsj < t p
j < tsj+1. Therefore γ is partially, but not eventually, imprisoned in U.
From definitions 8.2.6 and 8.2.7 we can conclude that if γ is partially imprisoned then either |Wγ|= 1 or |Wγ|=|R|. From proposition 8.2.14 and corollary 8.2.10 we
know, however, that if|Wγ|= 1 then γ is eventually imprisoned in all compact sets
K so that K∩W 6=∅. Whereas if|Wγ|=|R| then there exists a compact setK so
that K∩W 6=∅, but γ is not eventually imprisoned in K. Using proposition A.6.4 we can prove the following result.
8.2.15 Proposition. If γ is a future directed, precompact, winding, causal curve then the distinguishing condition fails to hold on γ.
Proof. As γ is winding, by proposition 8.2.4, |W| = |R| and therefore γ is inex- tendible. Since γ is compact and γ is totally imprisoned in γ, by proposition A.6.4 (proposition 6.4.8 of [59]), we see that the distinguishing condition cannot hold on
γ.
8.2.6
Overview
properties with regards to extendibility, precompactness and imprisonment. We summarise this in figure 8.1.