Nonstandard Analysis in Topology: Nonstandard and Standard Compactifications

0.12 MB·English

by  Sergio Salbany

Nonstandard Analysis in Topology: Nonstandard and Standard
Compactifications
by
S. Salbany and T. Todorov
6
0
0 Dedicated to Horst Herrlich on the occasion of his sixtieth anniversary.
2

n
Abstract Let (X,T) be a topological space, and ∗X a non–standard ex-
a
J tension of X. There is a natural “standard” topology ST on ∗X generated by
0 ∗G, where G ∈ T. The topological space ∗X,ST will be used to study, in a
3 systematic way, compactifications of (X,T(cid:0)). (cid:1)

]
N Mathematics Subject Classification: 031105,54J05, 54D35, 54D60.
Key words and phrases: non–standard extension, standard topology, com-
G
pactifications, monads.
.
h
t
a
1. Introduction
m
Let (X,T) be a topological space and ∗X,ST a non–standard enlarge-
[
ment, where ST is generated by ∗G, w(cid:0)here G ∈(cid:1) T. The space ∗X,ST
1
hasmanyinterestingtopologicalfeatures,but,asmaybeexpecte(cid:0)d,isvery(cid:1)
v
poorly separated as far as points are concerned.
4
2
7
We showthatawideclassofcompactificationsof(X,T)maybeobtained
1 by rendering ∗X,ST “separated”, thus illustrating the usefulness and
0
6 effectiveness, (cid:0)and broa(cid:1)d applicability of the non–standard compactifica-
0 tion ∗X,ST .
h/ (cid:0) (cid:1)
t Conceptually, it is not very common to regarda non–standardmodel ∗X
a
m asatopologicalspace,althoughthishasbeendone: A.Robinson[16]and
[17], W.A.J. Luxemburg [23],H. Gonshor [5],L. Haddad [6]. The ∗–open
:
v sets have always been part of non–standard techniques , but their role is
i
X more often at the level of the application of transfer principles than as
basic open sets of a topological space ∗X,ST .
r
a (cid:0) (cid:1)
In this paper we shall study the relationship between topological prop-
erties of (X,T) and their counterparts in ∗X,ST . This has led to a
unification and, perhaps, simplification of th(cid:0)e expos(cid:1)ition concerning com-
pactifications ([22], [14], [15], [23], [5], [6], [8]).
Arising from considerations related to the Theory of Frames, as well as
fromaninterestincompactificationsthatarerelevantto theoreticalcom-
puterscience,therehasbeenanincreasinginterestinT0–compactifications,
described as “well compacted” ([20]) and “stably compact” ([21]). We
1
shall show that these compactifications can also be obtained from non–
standardcompactificationsin a canonicalway. The relevantreference for
Frames(andalsostablecompactness)isP.Johnstone’sbookStoneSpaces
[10].
A methodological note is appropriate at this stage, concerning the role
of the axiom of choice – the axiom is essential in topology, to yield the
Stone-CˇechcompactificationofTychonoffspaces[12];itisalsoessentialin
thenon–standardapproachbyprovidingnon–standardenlargementswith
adequatesaturation([8],observationbeforeL ˘osTheorem4.5,ChapterII).
For the topological notions and constructs, we refer to J.L. Kelley [12],
and L. Gillman and M. Jerison [4]. For the relation between standard
andnon–standardmethods intopology,we referto L.Haddad[6]; for ba-
sic concepts, methods and further developments we refer to T. Lindstrøm
[13]. Forthe notionsconcerningcategorytheory,inparticularreflections,
we refer to [1], as well as [10].
2. Non–standard compactifications
∗
For any topological space (X,T) there is an enlargement X which is
saturated in the sense that if {F ⊆X|i∈I} is a family with the finite
∗ ∗
intersection property, then there is α in X which is in every F , i ∈ I
i
(see,forexample,[8],ChapterII,§8). Thus, ∗X,ST is acompacttopo-
logical space. (cid:0) (cid:1)
Thesets∗G,G∈T,constituteabaseforST. Forreference,wenotethat,
for A,B ⊆X, we have:
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
(i) φ = φ, X = X, (ii) (A∪B) = A∪ B, (iii) (A∩B) = A∩
∗ ∗ ∗
B;(iv) X − A= (X −A).
∗
The monad of x is µ(x) = { G|G∈T,x∈G}. More generally, for
∗
α∈ X, T
∗ ∗
µ(α) = { G|G∈T,α∈ G}, similarly we may define µ(A), where
∗
A⊆ X. T
We recall A. Robinson’s celebrated criterion for compactness: (X,T) is
∗
compact if and only if X = {µ(x)|x∈X}.
S
In general ∗X,ST is not a Hausdorff Topological space – the standard
∗
open sets a(cid:0)re inade(cid:1)quate to separate the rich assortment of points in X.
We shall provide and example, so as give an idea of the topologicalstruc-
ture of ∗X,ST .
(cid:0) (cid:1)
2
2.1 Example Consider N = {1,2,3,...} with the topology of upper
semi–continuity,basicopensetsbeingoftheformG ={1,2,...,n}, n∈
n
N. Let α ∈∗ N, then α ∈∗ G if and only if G ∈ α, hence α must
n n
be a principal ultrafilter since G is finite. Thus, every α for which
n
{H ⊆N|α∈∗ H} is a free ultrafilter, necessarily has only one ST–
neighbourhood, ∗N itself. Thus, no “non standard” α’s can be sep-
arated by ST–open sets.
Thefollowingobservationsareimportantbecausetheygivethefunc-
∗
toriality of the –extension.
2.2 Proposition Let (X,T) and (X′,T′) be topological spaces and
∗X,ST , ∗X′,ST′ non–standard compactifications. Then the
′ ′ ∗
(cid:0)function(cid:1)f (cid:0): (X,T) (cid:1)→ (X ,T ) is continuous if and only if f :
∗X,ST → ∗X′,ST′ is continuous.
(cid:0) (cid:1) (cid:0) (cid:1)
Proof For any G′ ∈ T′, we have that ∗G′ is a basic ∗T′ open set.
∗ ← ∗ ′ ∗ ← ′
Now ( f) [ G]=( f) [G] and the result follows.
2.3 Proposition Let f : (X,T) → (X′,J′), g : (X′,T′) → (X′′,T′′),
then ∗(g◦f)= ∗g◦∗f. Also ∗(IX)=I∗X.
We have indicated why ∗X,ST is a compact topologicalspace. In
fact,moreistrue. Firstl(cid:0)ysomet(cid:1)opologicaldefinitionsandtheirnon–
standard description.
2.4 Definition A topological space (X,T), not necessarily Hausdorff,
is calledlocally compact if for everyx∈X andopenV ∈T which
is a neighbourhood of x, there is a compact neighbourhood of x, W,
not necessarily open, such that W ⊆V.
Asimplenon–standarddescriptionoflocalcompactnessfollowsfrom
A. Robinson’s compactness theorem mentioned above.
Non–standard Local compactness: Forevery neighbourhoodV
of a given x∈X, there is a neighbourhood of x, W, such that
∗ ∗
W ⊆ {µ(x)|x∈W}⊆ V.
[
The following notion has been called supersoberness ([3], Chapter
VII, 1.10 Definition). When applied to a compact space (X,T), be-
cause it implies a precise form of compactness which specifies, not
only that ultrafilters have adherences, but that these should be of a
specialform,wehavetakenthelibertyofnamingitsupercompact-
ness. In [3], examples illustrating the usefulness of supersoberness
3
may be found in Chapter VII.
2.5 Definition Let(X,T)beatopologicalspace,notnecessarilyT0. (X,T)
is supercompact if for every ultrafilter U on X there is an essen-
tially unique point x such that the set of cluster points of U, adh U,
is the closure of the point x:
adhU = clX.
T
′
Thatxisessentiallyuniquemeans: ifx isanyotherpointwiththat
′
sameproperty,thenxandx havepreciselythesameT–neighbourhoods,
′
i.e. µ(x)=µ(x).
2.6 Examples
1. Every compact Hausdorff space is locally compact and super-
compact.
2. Every supercompact T1 space is compact Hausdorff.
3. Consider N∞ =N∪{∞}, where basic u–opensets in N∞ one of
the form G ={0,1,...,n} (thus, the only u–neighbourhood of
n
thepointatinfinityisN∞). Then(N∞,u)isaT0locallycompact
supercompactspace. Indeed,itisthe T0–locallycompactsuper-
compactcompactificationβ2(N,u)of(N,u),seeProposition4.4.
2.7 Theorem ∗X,ST isalocallycompact,supercompactenlargement
of (X,T). (cid:0) (cid:1)
Proof Wefirstestablishlocalcompactness,byshowingthat,forG∈
T,wehavethat∗GisST–compact. ConsiderafilterF ofclosedsets
∗ ∗ ∗
F , i∈I suchthat F ∩ G6=φ,alliinI. ThenF ={F ∩G|i∈I}
i i i
is a family of subsets of X which is closed under finite intersections.
Let U be an ultrafilter on X which contains F. By saturation,
∗ ∗
there exists p ∈ X such that p ∈ { (F ∩G)|i∈I}, i.e. p ∈
i
∗ ∗ ∗ ∗
( { Fi|i∈I})∩ G. Thus, p is a cluTster point of { Fi∩ G|i∈I}
∗
aTndbelongsto G,asrequired. Toprovesupercompactness,letV be
anultrafilteron∗X. NotethatadhV = H¯ H ⊆∗ X, H ∈V ,so
∗ ∗
that adhV = { F|F ⊆X, F is closedT, a(cid:8)nd(cid:12)F ∈V}. It is re(cid:9)adily
∗ (cid:12)
verified that Tp = {A⊆X| A∈V} is an ultrafilter on X. Hence
p∈∗ X. We show that adhV is the ST–closure of p, thus exhibiting
quite explicitly the special minimal point in the adherence. Firstly,
∗
p is in the adherence of V, since, given G∈T with p ∈ G, we have
∗ ∗
G∈V,by definitionof p. Hence Gintersects everyclosedsetin V
showingthatp∈adhV. Letα∈adhV. Ifαis notinthe ST–closure
∗ ∗
of p, then there is G ∈ T such that a ∈ G and p ∈/ G. But then
∗ ∗ ∗ ∗
p ∈ (X−G), hence (X −G) ∈ V, so that G∩ (X−G) = φ,
contradicting the fact that α∈adhV.
4
In Summary – Every topological space (X,T) may be embedded
intoasupercompactlocallycompactenlargement ∗X,ST withem-
bedding map η :(X,T)→ ∗X,ST . The assign(cid:0)ment is(cid:1)functorial
X
andηprovides,then,anatur(cid:0)altransf(cid:1)ormationfromtheidentityto∗.
It is natural, and important, to determine the behaviour of ∗ on
spaces that already compact. We shallexamine two cases: the clas-
sical case, even in the non–standard sense, when (X,T) is compact
Hausdorffandthe casewhere(X,T)is alocallycompact,supercom-
pact T0 space.
Beforewedoso,weshallexaminefurthersomeseparationproperties
of ∗X,ST .
(cid:0) (cid:1)
3. Non–standard compactifications and separation properties.
We shall describe conditions under which ∗X,ST is normal, or regular,
or a T0–space, in order to illustrate the na(cid:0)ture of S(cid:1)T on ∗X.
3.1 Proposition ∗X,ST is normal if and only if (X,T) is normal.
(cid:0) (cid:1)
Proof Assume that ∗X,ST is normal and let F1,F2 be disjoint
∗ ∗
closed sets of (X,T).(cid:0) Then (cid:1)F1 and F2 are disjoint closed sets
of ∗X,ST so, by assumption, they can be included in disjoint
ST–(cid:0)open se(cid:1)ts with disjoint closures. Restricting the open sets to
X provides two T–open sets G1,G2 with disjoint T–closures con-
taining F1 and F2, respectively. Conversely, assume (X,T) is nor-
mal. Let A,B be disjoint closed sets in ∗X,ST . By assumption,
∗ ∗ ∗ ∗
A = { F|F closed F ⊇A}, B = (cid:0){ H|H(cid:1) closed, H ⊇B}.
∗ ∗ ∗ ∗
Now FT∩ H =φforsome F ⊇A, H ⊇TB,whereF,H areclosedin
∗ ∗ ∗
X,otherwisethereisαin X suchthatα∈ (F ∩H),forall F ⊇A,
∗
H ⊇B. Thiswouldmeanthatα∈A∩B,whichisimpossible. Thus
we have F ∩H = φ so there are two disjoint open sets G1,G2 such
∗ ∗ ∗ ∗
that F ⊆ G1, H ⊆ G2. Then A ⊆ F ⊆ G1, B ⊆ H ⊆ G2 and
∗ ∗
G1∩ G2 =φ, as required.
3.2 Proposition ∗X,ST is regular if and only if every open set in
(X,T) is closed.(cid:0) (cid:1)
Proof Suppose (X,T) has the stated property and that α∈∗ G for
some G ∈ T. Since G is open and closed, so is ∗G, so ∗X,ST is
regular. Conversely,assume ∗X,ST is regularandletG(cid:0)∈T. I(cid:1)fG
werenotclosed,thenthereis(cid:0)x∈clG(cid:1)−G. Foreachopenneighbour-
T
hoodofx, H,wehaveH∩G6=φ,sothefamily{H ∩G|H ∈N }has
x
5
∗
the finite intersection property. By saturation, there is p∈ X such
∗ ∗ ∗
that p ∈ (H ∩G), for all H ∈ N . Thus p ∈ ( H ∩ G), hence
x
∗ ∗
p ∈ { H|H ∈Nx} G. By regularity, thereTis U ∈ T such that
∗T ∗ ∗ T ∗ ∗
p∈ U ⊆ cl U ⊆ G. ButthenX−clU = X − cl U ∩X ∈N ,
x
ST T (cid:18) ST (cid:19)
since x∈clG−G. Letting H =X−clU, we haveU∩H =φ, hence
T T
∗ ∗ ∗ ∗
H ∩ H = φ, which contradicts the fact that p ∈ U, and p ∈ H
∗
(since p∈ W for all W ∈N ).
x
3.3 Corollary 1 ∗X,ST is a completely regular space (no T0 separa-
tion assumed)(cid:0)if and on(cid:1)ly if every open set in T is closed.
Proof If ∗X,ST is completely regular, then it is regular, hence
(X,T) has(cid:0)the desi(cid:1)red property. Conversely, assume every open set
∗ ∗
is closed. Let G ∈ T and α ∈ X be such that α ∈ G. Because
∗
G is open and closed, there is a continuous real valued function
f : ∗X,ST → (R,m), where m denotes the usual topology, such
← ←
that(cid:0)G=f (cid:1)[0], X −G=f [1], the proof is complete.
3.4 Corollary 2 Let D denote the discrete topology on N. ∗N,SD is
not a T0–space. (cid:0) (cid:1)
Proof If ∗N,SD were T0, then it would be T2 since ∗N,SD
is regular, b(cid:0)y above(cid:1). In which case, since every bounded(cid:0)continu(cid:1)-
ous real valued function on (N,D) admits an extension to ∗N,SD
and (N,D) is dense in ∗N,SD (trivially, (X,T) is alwa(cid:0)ys dens(cid:1)e
in ∗X,ST , since ∗G ∩(cid:0)X = G(cid:1)), it would follow that ∗N,SD
is t(cid:0)he Ston(cid:1)e–Cˇech compactification β(N,D) of (N,D). I(cid:0)t is wel(cid:1)l
knownthatthisisimpossible(seeA.Robinson[16]page582;orK.D.
Stroyan and W.A.J. Luxemburg [23], 8.1.6, 8.1.7, 9.1); for a topolo-
gist, perhaps the easiest way is see this is to note that β(N×N) 6=
βN×βN (see, for example, [4]), whereas ∗(N×N)=∗ N×∗N.
3.5 Corollary 3 There is no topology T on N for which ∗N,ST is a
T0–space. (cid:0) (cid:1)
Proof Suppose the contrary, that ∗N,ST is a T0–space for some
topology T on N. The identity ma(cid:0)p i: ((cid:1)N,D) → (N,T) is con-
tinuous, hence so is its non–standard extension ∗i : ∗N,SD →
∗N,ST . Since ∗i is injective and ∗N,ST is T0, it(cid:0)follows(cid:1)that
(cid:0)∗N,SD(cid:1) is T0, which we know is imp(cid:0)ossible.(cid:1)
(cid:0) (cid:1)
3.6 Proposition ∗X,ST is a T0 space if and only if X is finite.
Proof IfX we(cid:0)reinfinit(cid:1)eand ∗X,ST aT0–space,thentherewould
be acountablesubsetN ofX(cid:0)withits(cid:1)relativetopology,alsodenoted
by T, giving ∗N,ST ⊆ ∗X,ST . Thus ∗N,ST is a T0 space,
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
6
which is impossible.
3.7 Corollary ∗X,ST is a Hausdorff space if and only if (X,T) is a
finite discrete(cid:0)space.(cid:1)
4. Non–standardcompactificationsofcompactspacesandstandard
compactifications
We shall first discuss briefly the compact Hausdorff case.
4.1 Proposition Let(X,T)beacompactHausdorffspaceand ∗X,ST
anon–standardextensionof(X,T). Thereisacontinuousr(cid:0)etraction(cid:1)
r : ∗X,ST →(X,T), with r being the identity when restricted
X X
to X(cid:0). (cid:1)
Proof By Robinson’s characterization, ∗X = {µ(x)|x∈X}.
∗
Thus,givenα∈ X,thereisxsuchthatα∈µ(x).SSince(X,T)isa
′ ′
Hausdorffspace,ifx6=x,wehaveµ(x)∩µ(x)=φ,henceα∈µ(x)
for a unique x. Define r (α) to be that x. Clearly, r (x) = x for
X X
∗
all x in X. If G ∈ T and x ∈ G, then α ∈ µ(x) gives α ∈ H for
∗
all H ∈ T that contain x. In particular α ∈ H, where H ∈ T is
such that x ∈ H ⊆ H¯ ⊆ G, (H exists by regularity of (X,T)). If
β ∈∗ H, and r (β) = x′, then x′ ∈ H¯, otherwise X −H¯ is an open
X
setcontainingx′,hence, bydefinition of r, ∗ X −H¯ =∗ X−∗ H¯
X
containsβ,whichisimpossiblesinceβ ∈∗ H. (cid:0)Thus r(cid:1)isacontinu(cid:0)ou(cid:1)s
X
retraction, as stated.
We now show that the Stone–Cˇech compactification of a Tychonoff
space (X,T) is simply ∗X,ST made Hausdorff. More pre-
(cid:16) (cid:17)
cisely,let[(X,T)]denote the T2–reflectionof(X,T)([1])then βX =
∗X,ST .
(cid:2)(cid:0) (cid:1)(cid:3)
4.2 Theorem Let(X,T)beaTychonoffspace.Thenβ(X,T)= ∗X,ST .
(cid:2)(cid:0) (cid:1)(cid:3)
Proof (X,T) is dense in ∗X,ST and the reflection map ϕ :
X
∗X,ST → ∗X,ST is (cid:0)continuo(cid:1)us, so ϕ (X) is dense in the
X
c(cid:0)ompact(cid:1)Haus(cid:2)d(cid:0)orff spa(cid:1)c(cid:3)e ∗X,ST . Consider f :(X,T)→(K,S)
where (K,S) is a compact(cid:2)(cid:0)Hausdor(cid:1)ff(cid:3) space. We show that there is a
map F, necessarily unique, such that F : ∗X,ST → (K,S) and
F ◦ϕ ◦η = f, where η : (X,T) → ∗(cid:2)X(cid:0),ST i(cid:1)s(cid:3)the embedding
X X X
mapof(X,T)intothenon–standardcomp(cid:0)actificat(cid:1)ion ∗X,ST . The
result then follows. (cid:0) (cid:1)
7
Naturality,inthecategoricalsense,oftheconstructsisbestexpressed
asa commutativediagram,givenbelow,fromwhichonecanreadoff
the required F. For convenience, we write [X] in place of [(X,T)],
and [f] for the reflected map.
η
X
(X,T) ∗X,ST
ϕX ϕ∗X (cid:0) (cid:1)
[η ]
X
∗
[X] [ X]
∗ ∗
f [f] [ f] f
[r ]
K
∗
[K] [ K]
[η ]
K
ϕK ϕ∗K
←
ϕ
K
r
K
∗ ∗
(K,S) ( K, S)
η
k
←
Observethatϕ hasaninverseϕ ,since(K,S)isalreadyHausdorff.
K K
← ∗
The required map is: F =ϕ ◦[r ]◦[ f], since
K K
← ∗
F ◦ϕ ◦η =ϕ ◦[r ]◦([ f]·ϕ )·η
X X K K X X
← ∗
=ϕK ◦([rK]·ϕ∗K)◦( f ◦ηX)
←
=ϕ ◦ϕ ◦r ◦η ◦f
K K K K
=I ◦I ◦f =f.
K K
We now consider T0 locally compact supercompact spaces.
4.3 Proposition Let (X,T) be a T0 locally compact supercompact
space and ∗X,ST a non–standard compactification. There is a
continuous(cid:0)retractio(cid:1)n r : ∗X,ST →(X,T), which is the identity
X
when restricted to X. (cid:0) (cid:1)
Proof Let α ∈∗ X. α determines an ultrafilter U on X. As
α
∗
usual, U = {A⊆X|α∈ A}. By supercompactness, there is x
α
suchthatadhU =clx. Define r (α) to be that x, whichis, infact,
α X
T
unique, as (X,T) is a T0 topological space. To prove continuity of
r : ∗X,ST → (X,T), consider α ∈∗ X and x = r (α). Given
X X
(cid:0) (cid:1)
8
V ∈ T with x ∈ V, local compactness ensures that there is W ∈ T
∗
and K compact such that x ∈ W ⊆ K ⊆ V. Now α ∈ W, other-
∗
wise α∈ (X −W), so that X −W ∈U , contradicting x∈adhU .
α α
∗ ′′ ′′
Consider now β ∈ W. We have, then, that β ∈µ(x ), for some x
∗ ′′ ′′ ∗ ′′
in K, since W ⊆ ∪{µ(x )|x ∈K} ⊆ V. Hence x ∈ adhU , so
β
′′ ′ ′ ′′ ′′ ∗
that x ∈cl x, where x =r (β). Now x ∈V, since µ(x )⊆ V,
T X
hence x′ ∈G, since x′′ ∈cl x′,provingthatr :(X,T)→ ∗X,ST
T X
is continuous. (cid:0) (cid:1)
As in the compact Hausdorff case, an entirely analogous proof will
showthattheT0locallycompactsupercompactreflectionof(X,T), β2(X,T),
is “ ∗X,ST made T0”. More precisely, let [ ]0 denote the T0–
(cid:16) (cid:17)
reflector, we have:
4.4 Proposition Let(X,T)beaT0–space. Thenβ2(X,T)= ∗X,ST 0.
(cid:2)(cid:0) (cid:1)(cid:3)
There is a note of warning that should be mentioned here – The
notion of“reflection” that is relevantis the notion of weak reflection
of H. Herrlich, and does not require the uniqueness of the map that
solves the extension problem: A subcategory R of a category A is
reflective, if for every A in A, there is R = R(A) in R and η :
A
A→R(A) such that if A′ ∈A and f :A→A′, then there is F, not
′
necessarily unique, such that F :R(A)→A and
F ◦η =f.
A
5. All Compactifications
Firstly,abriefreferencetocompactHausdorffcompactificationsofagiven
Tychonoffspace(X,T). Thesemayallbeobtainedbyauniformmethod,
as describedin([8], page158). The method givenabovedoes notreferto
continuous real valued functions on (X,T). However the T2 reflection of
∗X,ST can be seen to be induced by the ∗f′s where f ranges through
(cid:0)the boun(cid:1)ded continuousrealvaluedfunction, thus establishing arelation-
ship between the two approaches.
ObtainingallT0–locallycompactsupercompactificationscanalsobeachieved
by considering an analogue of the Q–compactifications referred to above
([8],page158)–oneconsidersfamiliesofcontinuousrealvaluedfunctions
into the Sierpin´ski dyad D={0,1},with Topology u={φ,{0},{0,1}}.
Lest the reader become too optimistic, it should be mentioned that it is
not possible to obtain all T0 compactifications of a T0 space as a type of
quotient of ∗X,ST – if it were possible, then the category of compact
T0 spaceswo(cid:0)uldbe ((cid:1)weakly)reflective in the categoryofT0 spaces,which
9
it is not, as shown by Miroslav Huˇsek ([9]; see also [7] for further devel-
opments) in response to a problem posed by Horst Herrlich.
Acknowledgement: The authors would like to thank the Scuola Inter-
nazional Superiore di Studi Avanzati, Trieste, for its hospitality and financial
support when this work was started.
The second author would like to thank Kent Morrison, Ed Glassco and Robert
Wolf for a critical reading of the manuscript and for the correction of many
errors.
The authors also wish to thank the referee for his comments, which have
broughtabout a substantial shortening of the originalpaper thus allowing it to
focus more clearly on its original intention – the effectiveness of non–standard
methods when dealing with compactifications.
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