classes 59
same argument can be used, in the reverse direction, to show that C(d0φ|φM×φM)⊂
C(dφ|φM×φM). Hence
dφ|φM×φM 'd0φ|φM×φM
and since ' is an equivalence relation we can conclude that
d'd0
as required.
These results, once combined can give us the following corollary which is a converse of 4.2.9.
4.2.11 Corollary. Let d be a distance on M and φ : M → Mφ a non-trivial
envelopment. If there exists a homeomorphism f :Md→φMso that f ı
d=φ, then
φ ∈E(d).
Proof. Let dφ be a complete distance on Mφ. Let d0 = dφ|φM×φM then, by def-
inition, φ ∈ E(d0) and so d0 ∈ D(M). From corollary 4.2.9 there must exist a homeomorphism h:Md0
→φMso that hıd0 =φ.
Let g : Md → Md0
be defined by g = h−1f. Then gı
d = h−1f ıd = h−1φ = ıd0.
Hence by proposition 4.2.8 we can see thatd 'd0. Therefore, from definition 4.2.5,
φ ∈E(d) as required.
4.3
A correspondence between the equivalence
classes
We begin with some definitions.
4.3.1 Definition. Let Φ be the set of all envelopments of M.
4.3.2 Definition. Let [φ] denote the equivalence class of φ ∈ Φ under the equiva- lence relation '.
4.3.3 Definition. Let [d] denote the equivalence class of d ∈ D(M) under the equivalence relation '.
We will show that there is a one-to-one correspondence between D('M) and Φ
' by con-
structing two functions which are inverses of each other. First we give the function from D('M) to 'Φ.
4.3.4 Definition. For each d ∈ D(M) choose φ ∈ E(d). We shall denote this chosen envelopment by φd. Define I :
D(M)
' →
Φ
' by letting I([d]) = [φd]. 4.3.5 Lemma. The function I is well defined.
Proof. Let d, d0 ∈ D(M) such that d ' d0. Since ϕd ∈E(d) and ϕd0 ∈ E(d0) there
exist two homeomorphisms h :Md→ ϕ
dMand f :Md
0
→ϕd0M. Also, as d' d0
there exists a homeomorphism g : Md → Md0
. Therefore f gh−1 : ϕ
dM → ϕd0M
is a homeomorphism. By construction f gh−1ϕd=ϕd0 and therefore by proposition
4.1.5 ϕd'ϕd0. Thus I([d]) =I([d0]) and I must be well defined.
Now we construct the function from Φ
' to
D(M)
' .
4.3.6 Definition. Let ψ ∈ Φ and choose d : Mψ × Mψ → R to be a complete
distance. Let dψ : M × M → R be defined by dψ(x, y) = d(ψ(x), ψ(y)), for all
x, y ∈ M. Note that by construction ψ ∈ E(dψ) and thus dψ ∈ D(M). Define
J : 'Φ → D('M) byJ([ψ]) = [dψ].
4.3.7 Lemma. The function J is well defined.
Proof. Letψ, φ ∈Φ such thatψ 'φ, then there exists a homeomorphismg :ψM → φM. Also, sinceψ ∈E(dψ) andφ∈E(dφ) there exist homeomorphismsh:Mdφ →
φM,f :Mdψ →ψM. Hence we have the homeomorphismf−1g−1h:Mdφ → Mdψ. By constructionf−1g−1hıdφ =ıdψ implyingdφ'dψ, by proposition 4.2.8. Therefore
J([ψ]) = J([φ]) so that J is well defined.
As presented I and J are dependent on a choice of an embedding and distance respectively. It turns out this choice is immaterial.
4.3.8 Lemma. For each d∈ D(M) the function I is independent of the choice of
φd.
Proof. Let d ∈ D(M), then in order to prove the result we need to show that if
ψ ∈E(d) then ψ 'φd.
As ψ ∈ E(d) we know, from corollary 4.2.9, that there exists a homeomorphism
f : Md → ψM so that f ı
4.3 A correspondence between the equivalence
classes 61
g : Md → φM so that gı
d = φd. Let h = f g−1, so that h : φM → ψM is
a homeomorphism. Also we know that hφd = f g−1φd = f ıd = ψ so that, by
proposition 4.1.5, we can see that ψ 'φd as required.
4.3.9 Lemma. For each φ ∈Φ the function J is independent of the choice of dφ.
Proof. Let φ : M → Mφ be an element of Φ then, in order to prove our result,
we need to show that for any two complete distances d : Mφ × Mφ → R and
d0 :Mφ× Mφ →R the induced distancesdφ:M × M →Randd0φ :M × M →R,
given bydφ(x, y) = d(φ(x), φ(y)) anddφ0(x, y) = d0(φ(x), φ(y)) (for allx, y ∈ M) are
equivalent. That is we need to show that C(dφ) =C(d0φ).
Let s ∈ C(dφ) then s is cauchy with respect to dφ and since d is complete, by
construction there must exist p ∈ ∂φM so that s → p uniquely. But this implies that s will be cauchy with respect to any distance onMφand therefore sis cauchy
with respect tod0. Hence, by construction,sis cauchy with respect tod0φ. Therefore
C(dφ)⊂ C(d0φ).
By similarity we can see that C(d0φ)⊂ C(dφ) and thusdφ'd0φ as required.
For clarity, we show that IJ = 1 andJI = 1 in two steps. 4.3.10 Lemma. Let d∈D(M), then JI([d]) = [d].
Proof. Let I([d]) = [φd], then φd ∈ E(d) so there exists a homeomorphism f :
Md → φ
dM. Since J([φd]) = [dφd] we know that there exists a homeomorphism
g : Mdφd
→φdM. Thus f−1g :Mdφd → Md is a homeomorphism. By consulting
the definitions we can see that f−1gıdφd =ıd and thereforedφd 'd. HenceJI([d]) = [dφd] = [d].
4.3.11 Lemma. Let ψ ∈Φthen IJ([ψ]) = [ψ].
Proof. Let J([ψ]) = [dψ] then there exists a homeomorphism f : Mdψ → ψM.
Let I([dψ]) = [φdψ] then there exists a homeomorphism g : M
dψ → φ dψM. Since f g−1 : φ dψM → ψM is a homeomorphism and as f g −1φ dψ = ψ we conclude that ψ 'φdψ. Therefore IJ([ψ]) = [φdψ] = [ψ]. Lastly, we have the main result of this chapter. 4.3.12 Theorem. The function I : D('M) → Φ
' is a bijective function with inverse J. That is, the sets D('M) and Φ' are in one-to-one correspondence with each other.
Proof. This follows from lemmas 4.3.10 and 4.3.11.
This theorem shows us that any information that can be extracted from 'Φ (e.g.
B(M)) can also be extracted from D('M). In particular, in chapter 6 we use the correspondence to reconstruct the Abstract Boundary using only D('M) and C(d). This gives us a new way to construct and think about the Abstract Boundary. Instead of thinking about boundary points in some envelopment, we can now think about collections of cauchy sequences with respect to some distance. In chapter 5 we use this new way of thinking to describe some algebraic structures on the set Σ0(M). In doing so we reformulate the correspondence of this chapter and show how to define an Abstract Boundary-like set for any topological space. By doing so we gain new tools to use in research and new ways to approach problems using the Abstract Boundary.