Embeddings and boundary sets
This subsection covers the necessary definitions and results for the construction of the Abstract Boundary. We begin by defining an envelopment, move on to boundary sets and lastly define an equivalence relation on boundary sets that will allow us to construct the Abstract Boundary.
3.1.1 Definition. Let M and M0 be manifolds of the same dimension. If there
exists φ :M → M0 a C∞ embedding, then M is said to be enveloped by M0, M0
is the enveloping manifold and φ is an envelopment. Since both manifolds have the same dimension, φMis open inM0.
3.1.2 Definition. A boundary point of an envelopment φ : M → Mφ is a point
p∈∂φM. A boundary set of an envelopment is a non-empty set B ⊂∂φM.
We are now in a position to define a partial order (the covering relation) on the set of all boundary sets of all envelopments. This partial order is used to construct the equivalence relation necessary to form the the Abstract Boundary.
3.1.3 Definition. Let φ : M → Mφ and ϕ : M → Mϕ be envelopments. Let
Bφ be a boundary set of φ and Bϕ a boundary set of ϕ. Then Bφ covers Bϕ or
equivalently BφBϕ if for every U ∈ Nφ(Bφ) there exists V ∈ Nϕ(Bϕ) so that,
φϕ−1(V ∩ϕM)⊂U.
When either of the boundary sets is a singleton {p}we shall write pB rather than
{p}B.
Before we give the equivalence relation, there are a number of important results about the covering relation that we shall need later.
3.1.4 Theorem. A boundary set B covers a boundary set B0 if and only if B covers every point p∈B0.
3.1.5 Theorem. A boundary set Bφ ⊂ ∂φM covers another boundary set Bϕ ⊂
∂ϕMif and only if for every sequence{xi} ⊂ M so that{ϕxi}has an accumulation
point in Bϕ, the sequence {φxi} has an accumulation point in Bφ.
Given the importance of this theorem in part II of this thesis, we provide an outline of the proof.
Proof. Assume that BφBϕ. Suppose that {xi} is a sequence in M so that {ϕxi}
has an accumulation point in Bϕ. Choose some nested collection of open sets
Ui ∈ Nφ(Bφ) so that
T
Ui = Bφ then using the covering condition it is possible
to construct a subsequence {yi} of {xi}so that {φxi}has an accumulation point in
3.1 The Abstract Boundary 27 Assume that Bφ 6Bϕ. Then there existsU ∈ Nφ(Bφ) so that for all V ∈ Nϕ(Bϕ),
φϕ−1(V ∩ϕM)\U 6=∅. This condition can be used to construct a sequence{xi}so
that {ϕxi}has an accumulation point in Bϕ but {φxi} has no accumulation points
in Bφ.
We are now able to define the necessary equivalence relation.
3.1.6 Definition. Given two boundary sets, B, B0, ifBB0 and B0B thenB is
equivalent to B0 and we write B ≡B0.
3.1.7 Theorem. The relation ≡ is an equivalence relation.
Curves
The classification of the boundary points requires the choice of a family of curves. We remind the reader of a few definitions and then define a property that we require our family of curves to satisfy.
3.1.8 Definition. A (parametrised) curve in a manifold M is a C1 function γ : [a, b) → M whose tangent vector γ0 nowhere vanishes. Such a curve will be said to start at γ(a). If b < ∞ the parameter is said to be bounded, otherwise the parameter is said to be unbounded.
For convenience we give the following definition, which was not used in [99], but is extremely useful.
3.1.9 Definition. Letγ : [a, b)→ M be a curve. A full sequence inγ is a sequence
{xi =γ(ti)}, with{ti} ⊂[a, b) a sequence so that so that i < j if and only if ti < tj
and limiti =b.
This definition allows us to state the next definition more succinctly than in the original paper. The concept of a full sequence is important in the Abstract Boundary, as will be shown in part III.
3.1.10 Definition. Let γ : [a, b) → M be a curve. A limit point of γ is a point
p∈ M such that there exists a full sequence {ti} inγ so that {γ(ti)} →p.
We now discuss the relationships between limit points of a curve and the partial order on boundary sets.
3.1.11 Definition. Let φ :M → Mφ be an envelopment. A curveγ : [a, b)→ M
approaches a boundary set B of φ if the curve φγ has a limit point in B.
3.1.12 Theorem. If a boundary set B covers a boundary set B0 then every curve in M which approaches B0 also approachesB.
Theorem 3.1.12 has no converse in the general case. We can, however, give a converse if we assume a connectedness condition.
3.1.13 Theorem. If every curve inMwhich approaches a boundary setBϕ ⊂∂ϕM
also approaches a boundary set Bφ ⊂∂φM, and if for all V ∈ Nφ(Bφ) there exists
U ∈ Nφ(Bφ) such that U ⊂V and φM\U is connected, then BφBϕ.
We continue the main development of this subsection.
3.1.14 Definition. An endpoint p∈ M of a curveγ is a point inM so that every full sequence in γ has p as a limit point. We write γ →p.
Equivalently, an endpoint p ∈ M of a curve γ : [0, b) → M is such that for all sequences {ti}i ⊂[0, b) so that {ti} →b, then {γ(ti)}i →p.
An implication of this is that the curve γ can be extended slightly to a curve λ : [0, b] → M by defining λ(t) = γ(t), ∀ t ∈ [0, b) and λ(b) = p, as long as limt→bλ
exists and is non-zero.
3.1.15 Definition. A curve λ : [c, d) → M is a subcurve of γ : [a, b) → M if [c, d)⊂[a, b) and λ=γ|[c,d). If a=c and d < b then γ is said to be an extension of
λ. In this case we say that λ is extendible. An inextendible curve is one that has no extension.
3.1.16 Definition. A change of parameter is a strictly monotone increasing C1
function, s : [a, b) → [c, d) so that s(a) = c. The curve λ : [c, d) → M is obtained from γ : [a, b)→ M by the change of parameter s if γ =λ◦s.
3.1.17 Definition. LetC be a family of parametrised curves in Msuch that,
1. for anyp∈ M there is at least one curve, γ ∈ C, passing throughp, 2. ifγ ∈ C then so is every subcurve of γ,
3. for any pair of curvesγ, λ∈ C which are obtained from each other by a change of parameter, either both parameters are bounded or both are unbounded.
3.1 The Abstract Boundary 29 Then C is said to satisfy the bounded parameter property (b.p.p.).
Examples of such collections of curves are; geodesics with affine parameter in a man- ifold, M, with affine connection, Cg(M); curves with generalised affine parameter in a manifold with affine connection, Cgap(M); timelike geodesics with proper time parameter in a Lorentzian manifold, Cgt(M).
3.1.18 Definition. A point p∈∂φM, where φ is an envelopment, is aC-boundary point or approachable if it is a limit point of some curve inC. Ifpis not approachable then we say that it is unapproachable.
The following is stated in [99] but not proved.
3.1.19 Lemma. If p≡q and pis a C-boundary point thenq is a C-boundary point.
Proof. Letp∈∂(φ(M)) andq ∈∂(ψ(M)) be boundary points so thatp≡q, where
φ : M → Mφ and ψ :M → Mψ are envelopments. Let γ ∈ C be a curve so that
p is a limit point ofφ(γ) then by theorem 3.1.12 we know that q is a limit point of
ψ(γ). Therefore q is a C-boundary point as required.