C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
E. L OWEN -C OLEBUNDERS R. L OWEN
M. N AUWELAERTS
The cartesian closed hull of the category of approach spaces
Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no4 (2001), p. 242-260
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© Andrée C. Ehresmann et les auteurs, 2001, tous droits réservés.
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
http://www.numdam.org/
THE CARTESIAN CLOSED HULL OF THE
CATEGORY OF APPROACH SPACES
by
E.LOWEN-COLEBUNDERS,
R LOWEN and M. NAUWELAERTSCAHIERS DE TOPOLOGIE ET
GEOMETRIE DIFFERENTIELLE CATEGORIQUES
Volume XLII-4 (2001)
RESUME. Cet article decrit le
plus petit 61argissement
cartesien ferme de lacat6gorie
des espacesd’approximation (’approach space’) AP,
c’est-a-dire
1’enveloppe
cart6sienne ferm6e deAP;
il est construit comme unesous-cat6gorie
des espaces depseudo-approximation
que deux desauteurs avait mdntr6 6tre le
quasi-topos topologique enveloppe
de AP.1. INTRODUCTION.
It needs no
argumentation
that cartesian closedness is animportant
and use-ful
property
fora category. Unfortunately
manycategories
are not cartesianclosed and therefore it is
interesting
to look for a cartesian closed modific-ation, especially
a so-called cartesian closedhull,
if it exists. Thus in[3], [ 11]
and[4], Antoine,
Machado and Bourdaud constructed the CCT hull ofTOP,
the category oftopological
spaces(and
continuousmaps)
and in[2],
Addmek and Reiterman constructed the CCT hull of
MET,
thecategory
ofmetric spaces
(and nonexpansive maps).
It is the purpose of this paper to construct the cartesian closed
topological
hull of
AP,
the category ofapproach
spaces(and contractions).
We dothis
by identifying
it with asubcategory
ofPsAP,
thecategory
ofpseudo- approach
spaces,which,
in[9],
was shown to be thequasitopos hull,
QTH(AP), of AP.
Remarkably,
whereas theobjects
of PsAP can be describedby
axiomsquite
similar to those
characterizing
theobjects
ofQTH(TOP),
the situation forCCTH(AP)
is somewhat different. Thesupplementary
axiomrequired
tocharacterize those
objects
of PsAP which are inCCTH(AP)
isirispired
not
only by
theapproach
of[11]
and[4]
for the case ofTOP,
but alsoby [2]
for the case of MET.2. PRELIMINARIES.
As will be
talking
about CCTcategories (or constructs),
first that243
a
well-fibred topological
c-construct in the sense of[1],
i.e. each structuredsource has an initial
lift,
every set carriesonly
a set of structures and each constant map(or empty map)
between twoobjects
is amorphism.
We willalways
assume a functor to be concrete(unless
this isclearly
not the casefrom its
definition)
andsubcategories (i.e. subconstructs)
to be full.We also recall that a construct A is CCT
(cartesian
closedtopological)
ifA is a
topological
construct which has canonicalfunction
spaces, i.e. for everypair (A, B)
ofA-objects
the sethom(A, B)
can besupplied
with thestructure of an
A-object,
denotedby [A, B],
such that(a)
the evaluation map ev : A x[A, B]
- B is anA-morphism, (b)
for eachA-object
C andA-morphism f :
A x C -->B,
the mapf * :
C -3[A, B]
definedby f *(c)(a)
=f (a, c)
is anA-morphism ( f *
is called the transpose off).
Note thatgiven f :
A x C ->B,
thetranspose f * :
C -3[A, B]
is the map which makes thefollowing diagram
commute:The CCT hull of a construct A
(shortly
denotedby CCTH(A)) (if
itexists)
is defined as the smallest CCT construct B in which A is closed under finite
products (see [7]).
Also from[7],
we recall thatgiven
a CCT construct C inwhich A is
finally
dense(i.e.
eachC-object
is a final lift of some structured sink inA),
the CCT hull of A is the full subconstruct of C determinedby CCTH(A)
:={C E C l
there exists an initial source(fi :
C-> [Ai, Bi])iEI
where Vi E I :
A;, B;
EA}.
In
short,
the CCT hull of A is the initial hull in C of thepower-objects
ofA-objects.
A more recent survey of such
properties
and hull concepts can be found in[6]
and[12].
Let us first introduce some notations in order to consider
CCTH(TOP)
asan
example (which
we shall beusing
lateron).
Given a setX, F(X)
standsfor the set of all filters on
X;
if 0 EF(X),
thenU(F)
stands for the set of all ultrafilters on X finer than Y. Inparticular, U(X)
:=U({X})
stands forthe set of all ultrafilters on X. Given A
c X,
we recall that stackand if A consists of a
single point
a, we also denote stack a := stack A. If 7 is a filter onX,
then the sec of 0 is defined assec
Now recall that
a pseudotopology (on X)
is a function q : X ->P (F(X))
assigning
to each xE X
a set of filters onX,
the filters"converging
tox", satisfying
theproperties:
: stack
The
pair (X, q)
is called apseudotopological
space and we also write F ->x
(or
even Fq -> x
orF x -> x)
instead of :F Eq(x).
A mapf : (X, qx)
-(Y, qy)
betweenpseudotopological
spaces is called continu-ous whenever
The construct of
pseudotopological
spaces and continuous maps is denotedby PsTOP,
which is atopological quasitopos
in which TOP can befinally
dense embedded
(hence
alsobireflectively) by associating
to eachtopolo- gical
space(X,T)
thepseudotopological
space(X, qT)
such tllat 7qT -> x
if and
only
if F -> x(in (X,T)) (i.e. V(x)
C1=).
It therefore follows from thegeneral theory previously
recalled that the CCT hull of TOP canbe found within
PsTOP,
the concretedescription
of whichrequires
thefollowing concepts.
Let
(X, q)
be apseudotopological
space. We denote its TOP-bireflectionby (X, q)
and define thepoint-operator (with respect
to(X, q))
asA
pseudotopological space X
is called an Antoine space orepi-topological
space if and
only
if it satisfies thefollowing
conditions(where
F° is the filtergenerated by {F° F
E7)
and lim 7 :={x
EX l F -> x}):
(1)
V7 EF(X) :
lim 7 is closed in(X, q) (closed-domainedness),
245
The full subconstruct of PSTOP
consisting
of Antoine spaces is denotedby EpiTOP
and it was shownby
work of A. Machado([11])
and G. Bourdaud([4])
thatEpiTOP
=CCTH(TOP).
Next,
we turn torecalling
some necessitiesregarding approach
spaces.An
approach
space is apair (X, b)
where X is a set and where S : X x2x
->[0, oo]
is a map, called adistance,
which has theproperties:
Given
approach
spaces(X, 6x )
and(Y, dY),
a mapf : (X, 6x )
-->(Y, Jy)
is called a contraction if it fulfills the
property
thatThe
category
AP ofapproach
spaces and contractions is atopological
con-struct and is
extensively
studied and described in[10].
In
[8]-[9],
it is shown that AP is a subconstruct ofPsAP,
thecategory
ofpseudo-approach
spaces, theobjects
of which are describedby
means of aconcept
of limits.A
map A : F(X) - [0, oo]x
is calleda pseudo-approach
limit(on X)
if itfulfills the
properties:
Note that
by (PsAL)
apseudo-approach
limit iscompletely
determinedby
its restriction to ultrafilters.
The
pair (X, A)
is called apseudo-approach
space. Givenpseudo-approach
spaces
(X, Ax)
and(Y, kY),
a mapf : (X, Ax)
-(Y, Ay)
is called acontraction if it fulfills the
property
thatThe
category
ofpseudo-approach
spaces and contractions is atopological
construct, denoted PSAP. For more information on this
category
we refer the reader to[9],
weonly
recall thoseproperties
which arerequired
in thesequel.
A
pseudo-approach
limit A :F(X) - [0, oo]x
is called anapproach
limitif it satisfies some additional
properties,
such as a so-calleddiagonal property (see
e.g.[10]),
which however we shall notrequire
to go intohere,
and whichensures it to be
equivalent
to a distance in the sense of thefollowing
resultwhich allows us to conclude that AP -> PsAP.
2.1. Theorem
([8]).
(1)
Given a set X and a distance S onX, As defined by
is an
approach
limit onX,
and vice versaif
A is anapproach
limiton X then
dk defined by
is a distance on
X,
such thatA8,
= k andd k d
= S.(2) If (X, dX)
and(Y, dY)
areAP-objects
andif f :
X -Y,
thenf
isa contraction
if and only if it fulfills
eitherof the following equivalent
conditions:
2.2.
Proposition.
(1) ([9])
Let(X, Ax)
and(Y, Ày)
bepseudo-approach
spaces, then thefollowing
areequivalent:
.
f : (X, Àx)
-(Y, Ay)
is a contraction.(2) ([8]-[9, proposition 5.7])
Let(Xi, Ài)ieI
be a classof PsAP objects.
If (fi :
X -(Xi, Ài))ieI
is a source, then the initiallift
A on X isgiven by
hence
(3) ([8]-[9, proposition 5.7])
Let(Xi, ;Bi)ieI again
be a classof PsAP
objects. If ( fi : (Xi, All)
-X)iEI
now is asink,
then thefinal lift
Aon X is obtained
as follows.
Given U EU(X)
and x EX,
we then. let
ifU
= stackotherwise
and
As we mentioned
earlier,
if we want to describe the CCT hull ofAP,
werequire
some CCT construct in which AP isfully
embedded. Thefollowing result,
shown in[9],
indicates that PSAP is agood
candidate.2.3.
Proposition.
PSAP is a cartesian closedtopological
construct. Moreprecisely, given PsAP-objects (X, Ax)
and(Y, Ay)
thepseudo-approach
limit A on
hom((X, Ax), (Y, Ay))
is determinedby
It is also shown in
[9]
that AP is bireflective in PSAP and we denote the bireflection of aPsAP-object (X, A) by (X, k). Furthermore,
PSTOPcan be
bireflectively bicoreflectively
embedded in PSAPby associating (X, Aq)
with aPsTOP-object (X, q),
where’xq(F)(x)
= 0 if Fq -> x
andAq(7)(z)
= o0 otherwise.Conversely,
the PsTOP-bicoreflection(X, qk)
of
(X, A)
is obtainedby letting F q k -> x
if andonly
ifk (F) (x)
= 0.Also TOP is
bireflectively bicoreflectively
embedded in AP([10]),
whereboth the
embedding
and the bicoreflection are restrictions of thepreviously
described functors.
Furthermore, given (X, S)
EAP,
one also finds that the closureoperator cls
of its TOP-bicoreflectionTd
satisfiescld(A) = {x
EX
l d(x , A)
=01, hence,
one finds itsneighbourhood
filters(V5(X))xEX by
Before we turn to the actual results and
constructions,
we recall some otherconcepts regarding approach
spaces which we will beusing
to obtain ourresults
(elegantly).
We will for instance bemaking
use of a characterization ofapproach
spacesby
means ofapproach systems.
First some
terminology.
A subsetA
of functions from agiven
set X to[0, oo]
is called an ideal in
[0, oo]x
if it is closed under theoperation
oftaking
finitesuprema, and if it is closed under the
operation
oftaking
smaller functions.Given A C
[0, oo]x
and afunction 0
E[0, oo]x,
then A is said to dominate0 if
If A contains
all 0
E[0, oo]x
that are dominatedby A,
then we say that Ais saturated.
A subset B of
[0, oo] x
is called an ideal basis in[0, oo]x
if for any a,B
EB,
there
exists,
E B such that aV B
-y.A collection of ideal bases
(g(x))xEx
in[0, oo]x
is called anapproach
basisif for all E X the
following properties
hold:(Bl) VO
E8(x) : O(x) =
0.(B2) Vo
EB (x), VE
>O,Vw
oo :3 (0z)zEx
En Li(z)
such thatzEX
If, additionally:
Vx
E X : B(x)
is a saturatedideal,
then
(B(x))xex
is called anapproach
system.Given a subset
B C [0, oo]x,
we defineB : =
10
E[0, oo] x 18
dominates01.
We call
B
the saturation of B.A collection of ideal bases
(B(X))xEX
is called anapproach
basisfor
anapproach
system(A(x)).,EX,
if for all xE X, A(x) equals
the saturation ofB(x).
In this case, we also say that(B(x))xEx
generates(A(x))xEX
or that(A(x))sEx
isgenerated by (B(x))xEX.
For ease in notation we may
also,
wheneverconvenient,
denote anapproach system (A(x))xeX simply by
A.We then have the
following
results from[10].
2.4.
Proposition.
(1)
Given a set X and a distance S onX, (Ad(x))xEX defined by
249
is an
approach
system onX,
and vice versaif (A(x))xEx
is an ap-proach
system onX, then dA defined by
is a distance on
X,
such thatAs,
= A anddAd
= d.(2) I!(B(x))xex
is anapproach basis,
then(6(X))-EX
is anapproach
system with
(B(x))xEx
as basis and(3) If (X, dX)
and(Y, dY)
areAP-objects
andif f :
X -->Y,
then thefollowing
areequivalent:
(i) f
is a contraction.In
(ii), if (AY(y))yEY
isreplaced by
anapproach
basisgenerating it,
the
equivalence
remains valid.In the
following,
we will also beneeding
aparticular AP-object
which wenow introduce. Let JP =
([0, 00], d P),
whereUsing
theforegoing,
oneeasily
finds thatTp
is theright
ordertopology
on[0, 00] corresponding
with the closureoperator cl,.
:=cldP,
i.e.Further,
that F converges tof3
in thistopology
is denotedby 7 r --> f3.
Since for every Jr on
[0, oo], lim r F
is closed inT,
we find thatlimr F =
[0, L (F)],
whereL(.F)
= suplimr F.
Recall the
following
usefulexpression
of£(0) given
in[9] (and
an easyconsequence).
2.5. Lemma. Let F be
a filter
on[0, oo],
then it holds that:and.
and then:
Some useful
properties:
2.6.
Proposition ([9]).
(1)
Let(X, d)
be anapproach
space, A C X and F EF(X),
thenc5( -, A) : (X, d)
--> Pand kd(F)(-) : (X, S)
--> JP are contrac- tions.(2)
P isinitially
dense in AP..3. CONSTRUCTION OF THE HULL.
3.1. Definition. Given
(X, À)
EPsAP,
wedefine
3.2. Definition. Given
(X, A)
EPsAP,
wedefine
where
OP
:=stack{GP l G
E91.
Recall that an extended
pseudo-quasi-metric (on X)
is a function d : X xX ->
[0, oo]
such that(i) d(x, x)
= 0(Vx
EX)
and(ii) d(x, z) d(x, y) + d(y, z) (Vx, y, z
EX)
and that thepair (X, d)
is then called an extendedpseudo-quasi-metric
space.3.3.
Proposition. Given (X, A)
EPsAP, (F(X), dX) is an extended pseudo-
quasi
metric space.Proof. Let
dx (F,9)
a anddX(9,H) (3.
Thisimplies
thatga
C J’and
it/3
Cg. Consequently, Ha+B C HB a
Cga C F.
Note that
dx
can attain the value oo,by
definition it isclearly
notsymmetric
and for instance
dx (Fo, 7)
= 0(hence,
it is notpossible
todrop
anyprefix
in the
previous proposition).
We
already
know that in anapproach
space, for any 0 EF(X),
the functionk (F) (-)
is a contraction. However we can also consider the function A withtwo
variables,
filters on X andpoints
of X. Theforegoing
definition of(F(X), dx)
now makes itpossible
to consider contraction and/orcontinuity properties
of this function of 2 variables.3.4. Definition. A
pseudo-approach
space(X, A)
is called anepi-approach
space
if
itadditionally satisfies the following
condition:The full
subconstructof PsAP consisting of epi-approach
spaces is denotedby EpiAP.
The
following
illustrateswhy
we could restrict ourselves to ultrafilters(as usual)
withoutcausing
any difficulties.3.5.
Proposition.
Given(X, A)
EPsAP, the following
areequivalent:
(1) (X, A)
EEpiAP.
(2) (X, A) satisfies
thefollowing condition:
(C)’ :
A :(F(X),7dx)
x(X, Tk) ([0, oo],T,)
is a continuous map.Proof. The
implication 2 => 1
is clear.To prove the
implication 1 => 2
assume thatA :
(U(X), 7dx) x (X, Ta)
-([0, oo], Tr )
is continuous(*).
Now let JF E
F(X)
and x E X such that KA(7)(z)
= supk(U)(x),
U E U (F)
hence,
there exists someultrafilter U D
F such thatÃ(U)(x)
> K.By (*),
we find V E
Vk(x)
and S > 0 such thatdX(u,W) d (where
W EU(X)) and y
E Vimplies
thatA (W) (y)
> K(**).
We now have to consider
some G
EF(X) and y
E X such thatdx (F, Q)
S and y E V. Since
g8
C :Fc U,
we find some W EU(G) : Wb
cLf.
Indeed,
assume otherwise that VW EU(Q),3W
E W :W d E
U.Hence
([10, proposition 1.2.2]),
we can findWl, ... , Wn
such thatWdi E u
(1
in)
andWI
U ... UWn
EG. However, since U D
F) 9d ) (W1
U ... UWn)’s
=Wd1
U ... UW,,s,
we find thatWdi
E U for some1 i n.
Consequently,
we obtained a contradiction and therefore wehave some W E
U (9) : Wd
CU, meaning dX(Zf,W)
S.By (**),
itfollows that
A(W)(y)
>K,
hence alsok(9)(y) > A(W)(y)
> K.Looking
forCCTH(AP)
caused thefollowing
niceproperty
to surface.3.6.
Proposition.
Let(X, A)
EEpiAP
anddX(F,9)
=0,
thenk(F) A(g). If in particular,
F =9°,
then evenk(F)
=k(9).
Proof. Let K
A(7)(z),
then we find a > 0 such thatdx(F,Q’)
a
implies
thatk(9’)(x)
>K,
hence, inparticular, A(g)(z)
> K. The arbitrariness of K nowimplies
thatk(F) A(G)-
As for the latter
claim,
since F =d° C 9,
we find thatk (9) k (F).
Con-sequently, combining
this with theprevious inequality,
we obtainA(7)
=k(9).
We are now in a
position
to state the main result.3.7. Theorem.
EpiAP
is the cartesian closedtopological
hullof
AP.We shall prove this in several
steps.
STEP 1: We first show that AP C
EpiAP.
3.8. Lemma. Let
(X, A)
be anapproach
space and let:F, 9
EF (X)
andp > 0,
then:Proof.
(1)
Let U Esecj:p,
we then claim thatU(P)
E sec Y.Indeed,
let F EY,
then we find zE U
such that also z EFp, meaning I(y, {z})
p for some y E F.Hence,
alsod (y, U)
p, i.e. y E F nU (p).
If we nowrecall from theorem 2.1 that
and
then the
foregoing clearly
demonstrates what wasrequired.
(2)
LetdX (F, 9)
a, hencego
C r.Consequently, by (1), k(F)
a(9a) A(g)
+ a.By
the arbitrariness of a, we conclude thatA(7) A (g)
+d x (F, 9).
The
foregoing
lemma shows that in anapproach
space, for any x EX,
alsothe function
a(-)(x)
is a contraction. However we can show more.3.9.
Proposition.
Let(X, A)
be anapproach
space and let(A(x))xex
be theapproach
system associated with À.If we
putthen
(B O)(F,x)EF(X)xX
is anapproach
basis onF(X)
x X andis a contraction.
Proot Let
(BP(x))
:={O
E[0,oo][o°°] lO dP(x,.)}
if x E[0, oo[
andlet
(Bp(°°))
:={8]a,oo] I 0
aoo} (where BA : [0, oo]
->[0, oo]
is suchthat
6A(A) = 101
and8A([0, oo] B A) = {oo}
for any subset A of[0, oo]).
Itthen follows from
proposition 2.4.(2)
that thisapproach
basis generates theapproach system
associated withSip.
0 w oo and 0 c be fixed. As
k(F) : (X, A)
-> P is a contraction(by proposition 2.6), proposition 2.4.(3)
allows us tofind 0
EA(x)
such thatWe then find that
Hence, by
the arbitrariness of w and E, this demonstrates what wasrequired (in
case of theassumption k (F)(x) oo).
Next we assume that
À(F)(x)
= oo,hence,
it now needs to be shown that forarbitrary
E > 0 and 0K, w
oo, thereexists 0
EA(x)
such thatVig
EF(X),Vy E X: 8]K,oo](k(9)(y))
^ wd X(F, Q)
+0(y)
+ f.As
A(0) : (X, À)
-> P is acontraction,
wefind 0
EA(x)
such thatWe now claim that
If
dx (F, Q)
> w ork(9)(y)
>K,
this isclearly satisfied,
so let us assumethat
d x (F, 9) w and k(9)(y)
K.By
theprevious lemma,
we then findthat
k(K)(y) A (g) (y)
+dx (F, G) K
+ w, hence3.10.
Corollary.
AP CEpiAP.
Proof.
Using
notations asbefore,
it iseasily
seen that the TOP-bicoreflection of(F(X)
xX, 8;)
is(F(X)
xX, 7d,
x7i) and, by definition, TP
=Tr.
STEP 2: Our next
goal
is to show thatEpiAP
is a cartesian closedtopolo-
gical
construct.3.11.
Proposition.
Letf : (X, Ax)
-(Y, Ay)
be a contraction betweenPsAP-objects,
thenf : (F(X), dx)
-->(F(Y),dY) :
F l->f(F)
is alsoa contraction.
Prool: As
f : (X, Ax)
-(Y, Ay)
is acontraction,
we find thatand, hence, f (F)°P
cf(FP) (for
all F EF(X) and p 2: 0). Thus, GP
C Fimplies
thatf(9)°p
Cf(QP)
Cf (F),
which means thatdY (f(F), f(9)) dx(F,9).
3.12.
Proposition. EpiAP is bireflective in PsAP, in particular, EpiAP
is a
topological
construct.Proof. Let
(fi : (X,A)
--->(Xi, ki))iEI
be initial inPsAP,
where all(Xi, ,xi)
eEpiAP.
To show that(C)
issatisfied,
assume that A E7;,
thenit follows from
2.2(2)
thatFrom the
contractivity
of allfi,
i EI,
theforegoing proposition
and the fact that allÀi,
i EI, satisfy (C)
it follows that À -1(A)
is open. Hence B is continuous and(X, k)
EEpiAP..
3.13.
Proposition.
Let(X, kX)
and(Y, kY)
bePsAP-objects,
and letg
bea filteron X,
then themap 9 : F(hom((X, Ax), (Y, Ay)))
-F (Y) : T
HY(9)
is a contraction, i.e.for
anypair (D
and Yof filters
onhom(X, Y):
dy (O(9) , Y (9)) dhom (x,y) (O, T) -
Proof. As evx :
[X, Y]
- Y(x
EX)
arecontractions,
we find thatand, hence, O (F)°P
COP(F) (for
all 0 EF(hom(X, Y))
and JF EF(X)).
Consequently, Y P
C (Dimplies
thatBI1(g).p
CY p(9)
CO(9),
which means255
3.14.
Proposition. EpiAP
is closedunder formation of power-objects
inPsAP. Moreover,
if (X, Ax)
E PSAP and(Y, Ay)
EEpiAP,
then[(X, AX), (Y, Ay)]
EEpiAP.
In
particular, EpiAP
is a cartesian closed category.Proof. Let
(Z, k)
:=[(X, AX), (Y, Ay)],
where(X, Ax)
E PSAP and(Y, Ay)
EEpiAP.
To show that(Z, A)
satisfies(C),
assume that A E7,,
then it follows from the formula of A
(see proposition 2.3)
thatHence it follows from the fact that all evx, x
E X,
arecontractions,
theforegoing proposition
and the fact thatAy
satisfies(C)
thatk-1 (A)
is open.Hence
(Z, A)
EEpiAP,..
STEP 3: We now turn to
showing
that proper"density"
conditions are sat-isfied.
Some of
this,
APbeing finally
dense inEpiAP,
hasalready
been con-sidered in a more
general
way in[9,
section3]
and we will recall the results needed herebriefly.
3.15.
Proposition.
APis fznally
dense inEpiAP.
Proof. As in
[9],
we first define thefollowing approach
spaces. Given a setX, 7
EF(X)
andf :
X -[0, oo],
we definek(F, f) : F(X)
-->X[0,°°]
by
7 n stack x C
9, 9 #
stack xg
= stack xF n
stack x ct. g.
It is then shown in
[9, proposition 3.1]
that(X, A(y,f))
is anapproach
space.If we now consider
(Z, A)
EEpiAP,
then oneeasily
finds(as
in[9,
pro-position 3.2])
thatis a final sink in
PsAP,
hence also a final sink inEpiAP. m
Assume without restriction in the
following
thatX # 0.
3.16.
Proposition.
Let(X, k)
EEpiAP, then j :
X -->[[(X, A), P], P]
defined by j( x )(f)
=f (x)
is an initial contraction.Proof. We first
give
thefollowing diagram
forclarity
and mention thatj
:=evil
is the map which makes thefollowing diagram
commute:Hence, by properties
ofpower-objects, j
is a contraction.In the
following
we also letand
To prove
that j
isinitial,
we will show for every ultrafilter U onX,
aE X
and 0 K oo thatk(u)(a)
> Kimplies
thatkHH(j(u))(j(a)) >
K.By proposition 2.2,
we then findthat j
is initial.It will be shown that
kHH(j(u))(j(a)) >
Iiby defining
anappropriate
g Ehom((X, k), P)
and BII EF(hom((X, Ax), P))
such thatkP(y(u))(g(a)) = À(j(U)(BI1))(j(a)(g)) 2::
K >kH(y)(g),
hence,by
thedescription
of func-tion spaces in
PsAP, (see proposition 2.3), kHH(j(u))(j(a)) >
K.Defining of 9
Ehom((X, k),P)
and B11 EF(hom((X, kX),P)):
By proposition
3.5 and the fact that(X, A)
EEpiAP,
we find V EVk(a)
and 6’ > 0 such that for all 7 on X with
dX(U, F)
8’ and x eV,
wehave k(F)(x)
> K(*).
Let go :
(X, A)
---> P : a -+dk(a, X - V) (which
is a contractionby proposition 2.6).
As V EVk(a),
we find thatd1
:=go(a)
> 0.First assume that
go(a) K,
then define gl := go +(K - Si)
andfinally
9 := gl A K and 6" :=
Si.
It followsthat g
Ehom(X, P), 9 K, g(a)
= Kand
{g
> K -6")
c V.If however
go(a) > K,
thendefine g
:= go ^ K and choose(any)
0 6"K, then g
and S" also fulfill theforegoing properties.
Now choose 0 S 6’ A S" and let B11’
{x}d E
257
contradiction.
Furthermore,
suchF’d
is never a void set, as italways
containsthe constant oo-function.
Also,
it holdsthat F1d
nP28
=(Fl
UF2)^d, (Fl
UF2).d
=F15
UF25
and Lf is anultrafilter,
hence w’ is a filterbasis. Now let W be the filter onhom(X, P) generated by
T’.Proving
thatkH(y)(g)
R’ - 6 K:By
thedescription
of function spacepseudo-approach
limits(see proposition 2.3),
it needs to be shown for alll xE X
and F EF(X)
thatTo this
end,
let xE X
and 0 EF(X)
be such that(I)
holds. Hence K - 6kP(y(F))(g(x)) g(x), consequently g(x) > K - d and so x E V (II).
We also find
that Fd C l.l (meaning dX(u, F)
d d’(111)).
d
Eu,
Indeed,
if this were not the case, then we could find F E 0 such thatFd E U,
implying Ps
EB11,
hence]S, oo]
EW(0)
and therefore inparticular, V/?
>K - d :
]g(x) - 0, oo]
Ey(F) (as g(x) K).
It then follows from lemma2.5.(2)
thatkP(y(F))(g(x))
K -6,
which contradicts(I).
Consequently,
it follows from(*) (and (II)
and(IIn)
thatk(F)(x)
> K.Also, kP(y(F))(g(x)) C g(x) K,
hencekP(y(F))(g)) A(0)(z).
Proving
thatkP(y(u))(g(a)) >
K:Let us assume the
contrary,
i.e.kP(y(u))(g(a)) K-E,
where 0 c K.This
implies
thatVB
> K - E :(g(a) - B,°°]
Ew(U),
henceVB
>K - E,
3U E U :ÛK-B
E y. Inparticular,
we have someI 2::
0 and UE u
such that
l/T
E y.Consequently, Fd C Û’Y
for someF # 0, Fd E
U.However, Ps
CÛy implies
that UC Fa (hence
acontradiction). Indeed,
let z V Fd, implying 6x(-, (z))
EPs,
hencedk(-,{z})
EÛ’Y,
thereforez EU. 5
STEP 4: Now we are in a
position
to combine allprevious
results and toprove the final
step.
3.17. Theorem.
EpiAP is the
cartesian closedtopological
hullof AP.
Proof. For this to be the case, we need
(as
notedbefore)
thatEpiAP
isa cartesian closed
topological
construct(which
has been verified instep 2)
and that AP is
finally
dense inEpiAP (which
was verified insteps
1 and3).
We also need that the classis
initially
dense inEpiAP.
However,by
theforegoing proposition,
forany
(X,.X)
EEpiAP
we have an initialmap j :
X -->[[(X, k), P], P]
and since the functor
[-, JP] : EpiAP --> EpiAP
transforms finalepi-
sinks into initial sources
(see [7,
lemma6]) (and by proposition 3.15,
we canobtain
[(X, k), P]
as a final lift of anepi-sink involving AP-objects),
we findthat H is indeed
initially
dense inEpiAP,..
4.
EpiLOGUE
We now show that
EpiTOP
=CCTH(TOP)
has a nice relation toEpiAP.
4.1.
Proposition.
Let(X, q)
EPsTOP,
then(X, Aq) = (X, Aq).
Proof. We will prove this
by showing
that(X, kq)
is also the AP-bireflection of(X, aQ).
To thisend,
letf : (X, kq)
-->(X, S)
be acontraction,
where(X, d)
E AP. But then alsof : (X, Aq)
->(X, qd)
is acontraction,
where we recall that the latter space is the PsTOP-bicoreflection of
(X, 6) .
Since we observed earlier that the TOP-bicoreflection in AP is
just
therestriction of the
PsTOP-bicoreflection,
it follows that(X, qs)
is atopolo- gical
space, hencef : (X, Aq)
->(X, qs)
is a contraction.Consequently, f : (X, kq)
->(X, S)
is a contraction. Thefollowing diagram
illustrates thisargumentation:
4.2.
Proposition. EpiAP
n PsTOP =EpiTOP.
I Proof. If
(X, A)
EPsTOP,
oneeasily
finds that condition(C)
isequival-
ent to
and Also observe that in this case
90 = g.
Now assume that
(C)t
holds(i.e. (X, À)
EEpiAP
nPsTOP).
Sincepo C F (for
all :F EF(X)),
we find that VY+-> a,3V
EV(x,7i)(a) :
259
(for
all 1-£ EF(X)). Consequently, (X, A)
is aclosed-domained, point- regular pseudotopological
space(i.e.
an Antoinespace).
Conversely,
assume(X, A)
EEpiTOP
and Q-/->
a. Let V := X -lim0 E
V(x,7i)(a) (as (X, À)
isclosed-domained)
and suppose00
C :Fand 2: E V. We then find
that -/->
x, for it this were not the case, thenalso
d°
-> x(as (X, A)
ispoint-regular), implying
F -> x, which is acontradiction. o
4.3.
Proposition. EpiTOP
isbireflective
andbicoreflective
inEpiAP.
Proof. The first claim is clear. As for the second
claim,
let(X, A)
EEpiAP,
then we show that(X, A’),
the PsTOP-bicoreflection of(X, À) belongs
toEpiTOP.
To this
end,
assume that Qf4
x, i.e.À(F)(x)
> 0. Since(X, A)
EEpiAP,
we find V EV(X,7i)
and 6 > 0 such that ford(x,k)(F, g)
S andy E
V,
we have thatA (9) (y)
> 0.As
Ix : (X, a’)
->(X, A)
is acontraction,
we find thatV
EV(x, Tk-,)
andthat
d(X,k’)(F, 9)
Simplies
thatd(X,k)(F, 9)
S(by proposition 3.11).
Putting together
what we have found sofar,
we obtain:VF qk’-/->x,ZVEV(X,Tk’) : (yEV^9°CF) => K(9)(y)>0 =>9 qk’-/->y.
Therefore, (X, A’)
satisfies(C)t
andbelongs
toPsTOP,
hence(X, A’)
EEpiTOP
The
foregoing
results are combined in thefollowing diagram:
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E. Lowen
Department
of MathematicsUniversity
of Brussels(VUB)
Pleinlaan
2,
1050Brussel, Belgium
R. Lowen and M. Nauwelaerts*