C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
D OUGLAS B RIDGES H AJIME I SHIHARA P ETER S CHUSTER L UMINITA V Î ¸ T ˘ A
Products in the category of apartness spaces
Cahiers de topologie et géométrie différentielle catégoriques, tome 46, n
o2 (2005), p. 139-153
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139
PRODUCTS IN THE CATEGORY OF APARTNESS SPACES
by Douglas BRIDGES, Hajime ISHIHARA,
Peter SCHUSTERand
Luminita Vi01630103
CA HIERS DE TOPOLOGIE ET
GEOMETRIE DIFFERENTIELLE CATEGORIQUES
Volume XL VI-2 (2005)
Resume. Les auteurs ddfinissent une structure de
separation
sur leproduit
de deux espaces deseparation (’apartness spaces’)
etanaly-
sent le role de la
decomposabilite
locale dans la th6orie. Tout le tra- vail est constructif - c’est-A-dire utilise unelogique
intuitionnisteplut6t
queclassique.
1 Introduction
The axiomatic constructive
theory
ofapartness
spaces,originally
introducedfor
point-set
apartness in[3],
was lifted to the set-setapartness
context in[7]
and
[9]. Among
the notable omissions from the latter paper was a notion ofproduct apartness structure, corresponding
in the set-set case to thepoint-
set case covered in
[3].
Werectify
that omission in thepresent
paper. We show that in the presence of a condition known as localdecorrtposability,
acondition that
automatically
holds under classicallogic
when theinequality
relation is the denial of
equality,
both thepoint-set
and the set-set notionsof
product
space arecategorical.
Note that in this paper we discussonly products
of two(and hence, by
asimple extension,
offinitely many)
apartnessspaces; we do not deal with
products
of infinite families.Our work is
entirely constructive,
in that we useonly
intuitionisticlogic.
We
expect
that most of it can beformalised,
with someeffort,
in a construc-tive
(and predicative)
settheory
such as is found in[1].
For aclassical-logic-
based
analogue
ofapartness
spaces see[6].
140
2 Point-set apartness
We work
throughout
with a set Xequipped
with abinary relation #
ofinequality,
orpoint-point apaTtness,
that satisfies theproperties
A subset S’ of X has two natural
complementary
subsets:. the
logical complement
O the
complement
Initially,
we are interested in the additional structureimposed
on Xby
arelation
apart (x, S)
betweenpoints x
E X and subsets S’ of X . For conve-nience we introduce the
apartness complement
of
S; and,
when A is also a subset ofX,
we write A - S inplace
of A n -S.We call X a
point-set apartness
space if thefollowing
axioms(intro-
duced in
[3])
are satisfied.A1 x #
y ->apart(x, {y})
A2
apart(x, A) -> x E
AA3
apart (x,
A UB) apart(x, A)
Aapart(x, B)
A4
(apart (x, A)
A -A C~B) -> apart(x, B)
A5
apart (x, A) -> Vy
E X(x f
y Vapart(y, A))
We note, for future
reference,
that it can be deduced from axioms A5 and A2 that-A c- A;
with this and theremaining axioms,
if we alsorequire
the
inequality
on X to benontrivial,
in the sense that there exist x and y in X with x#
y, then we can show that X =-0 (see [3],
Section2).
Notealso that if
apart (x, {y}), then x #
y.The
morphisms
in thecategory
ofpoint-set apartness
spaces X and Yare those functions
f :
X - Y that are continuous in the sense that141
The canonical
example
of apoint-set apartness
space is a metric space(X, p) ,
on which theinequality
andapartness
arerespectively
definedby
and
Functions between metric
point-set apartness
spaces arecontinuous,
in thesense defined
above,
if andonly
ifthey
are continuous in the usual C-J sense.The
metric-space example generalises
to uniform spaces, but we defer further discussion of those until we deal with set-setapartness
later. Anotherexample
of apoint-set apartness
isgiven by
aT, topological
space(X, r)
with a nontrivial
inequality #;
in this case, theapartness
is definedby
and we must
postulate
A5 since it need not hold ingeneral (see [3]).
Let
Xi
andX2
bepoint-set apartness
spaces, let X be their Cartesianproduct X,
xX2, and,
forexample,
let x denote the element(Xl, X2)
of X.The
inequality
relation on X is definedby
We define the
product point-set apartness
structure on X as follows:where -Uk
is the apartnesscomplement
ofUk
in thepoint-set apartness
space
Xk .
We callX, equipped
with thisapartness structure,
theproduct
of the
point-set apartness
spacesXl
andX2.
Such
product
spaces are discussed in[3],
where it is first verified that theproduct apartness
structurereally
satisfies Al-A5. Theproofs
of a numberof natural results on
product point-set apartness
spaces in[3] required
thespaces
Xi, X2 (equivalently,
theproduct
spaceX)
to becompletely regular, according
to a definition that we do not need here. We first state the mainproperties
ofproduct point-set apartness
spaces,replacing complete regular-
ity by
the much weakerproperty
of localdecomposability;
those results for which thisreplacement
is made areproved
in full.142
A
point-set apartness space X
is said to belocally decomposable
ifLocal
decomposability always
holdsclassically:
if x E-S,
then as X =-S’ U - - S and -S = -N -
S,
we can take T = ~ - S.Every
metricspace
(X, p)
islocally decomposable:
for if x E X -S, then, choosing
r > 0 such thatB (x, r)
C-S,
we can take T=~B (x, r/2)
to obtain x E -T andX = T U -S’. It also holds for a uniform space as defined in
[7].
Lemma 1 Let X -
X,
xX2 be
theproduct of
twopoint-set apartness
spaces, and let
Ai
CXi.
Then-AI
xX2 = - (AI
xX2)
andXl
x-A2 =
-
(Xl
xA2) .
Proof. Since
-A1
xX2
=-A1
x-0
andthe definition of the
product
apartness shows that-A1
xX2 C -(A1
xX2).
Conversely, given (Xl, X2)
in -(AI
xX2) ,
use the definition of theproduct apartness
to find subsetsUi
ofXi
such thatIf E E -U1,
thenso for all q E
A1
we have(ç, X2) i (n, X2)
andtherefore ç i
n]. Hence x1E -Ul C~A1.
It now follows from axiom A4 thatapart (xl, A1) .
Thus-
(AI
xX2)
c-A,
xX2.
The other
part
of the lemma isproved similarly.
Proposition
2 Let X= Xl
XX2
be theproduct of
two inhabitedpoint-set apartness
spaces. Then X islocally decomposable if
andonly if
eachXk
islocally decomposable.
Proof.
Suppose
first that X islocally decomposable.
Let xi E-Ul
CXl,
and
pick
X2 EX2.
Then143
Hence, by
definition of theapartness
onX,
x E -(Ul
xX2) . So, by
the localdecomposability
ofX,
there exists T C X such thatLet
and consider
any E
EXi.
Either(E, x2 ) E - ( U1
xX2 )
and therefore(by
Lemma
1) E E - Ul ,
or else(E, x2 ) E
T andso E
EVI.
It remains to show that x lE - Yl .
To thisend,
since xE -T,
we can findWi C Xi
such thatx E
-W1
x-W2
C~T. Forany E
E-W1
and v EV,
we have(E, x2)
E~Tand
(v, x2 ) E T; whence E# v .
Thus-w1C~V1.
Since x 1e -Wi,
it followsfrom axiom A4 that x,
E -Vl,
asrequired.
A similarproof
shows thatX2
islocally decomposable.
Now suppose,
conversely,
that eachXk
islocally decomposable.
Considerx E X and S C X such that x E - S. Choose
Ul
CX,
andU2
CX2
such that x E-Ul
x-U2
C -S. Since Xk E-Uk,
there existsVk C Xk
such thatLet
Then -V1 X -v2 ENT,
so, inparticular,
x E -T. On the otherhand,
foreach E
EX,
eitherE1 E -Ul
andE2 E -U2,
in whichcase E
E-S;
or else wehave either
E1
EVi
org2
EY2, when E
E T. Thus X = -S UT,
and so X islocally decomposable.
Corresponding
topoint-set apartness
there is theopposite property
ofnearness, defined
by
Classically,
if theinequality
is the denial ofequality,
thennear(x, A)
holdsif and
only
if-,apart (x, A) ,
which isequivalent
to the conditionTo discuss this
constructively,
we first observe that ifXi
is apoint-set apart-
ness space
and Ui
cXi (i
=1, 2) ,
then-144-
Proposition
3 Let X =X1
x-Y2
be aproduct of
twopoint-set apartness
spaces, let x be a
point of X,
and let A be a subset of X. Then the following conditions areequivalent.
(i) for
allU,
CXl
andU2
CX2
such that x E-Ul
x-U2,
there existsy E
(-ui
x-U2)
n A.(ii) near (x, A) .
Proof. For the
proof
that(i) implies (ii),
see[3]. Conversely,
ifU,
CXl, U2
CX2,
and xE -U1 x -U2,
thenby
thepreceding observation,
x E-
((U1
xX2 )
U(Xi
xU2)).
If also near(x, A) ,
then there exists y inwhich, again by
our observationabove, equals ( - Ul x- U2 )
n A. mProposition
4 Let X =Xl
xX2
be theproduct of
twopoint-set apartness
spaces, let x E
X,
and let A c X.Suppose
that thefollowing
condition holds.Then
apart (x, A) . Conversely, if
the spacesXl, X2
arelocally decomposable
and
apart (x, A) ,
then condition(*)
holds.Proof. First assume
(*),
and constructVI, V2
with the statedproperties.
To prove that
apart (x, A) ,
it suffices to note thatwhere in the first line we have used the observation
preceding Proposition
3.Now assume,
conversely,
that the spacesXl, X2
arelocally decomposable
and that
apart (x, A) .
ChooseUi C Xi
such that-145-
For each
i,
we use the localdecomposability
ofXi
tofind Vi
CXi
such thatxi E
-v
andXi = - Ui
UV,.
ThenFor if
(ÇI,Ç2) E A,
then eitherÇl E -U,
andÇ2 E -U2,
which isimpossible,
or
else,
as must be the case,61
EVI
or62 E V2.
Before
going
anyfurther,
we should proveLemma 5 Let
X1, X2
bepoint-set apartness
spaces. Then theprojection mappings
prk :Xl
xX2 -> Xk
are continuous.Proof. Let
apart (pr1 (x), pr1 (S)),
where S C X =Xl
xX2;
thenby Proposition
28 of[3],
there existsUl
CX1
such thatThen x E
-U,
xX2 = -U,
x-0. Also,
if y E-Ul
x-0,
then y1 E-Ul,
so for all z E
S’, y1 #
z1 and thereforey #
z.Thus - U1 X-0 c - S,
and therefore
apart (x, S) .
This proves thecontinuity
of prl; that of pr2 is establishedsimilarly.
We can now demonstrate the
categoricity
of theproduct
of twopoint-set apartness
spaces.Proposition
6 Let X =X,
XX2 be
theproduct of
twolocally decomposable point-set apartness
spaces, andf
amapping of
apoint-set apartness
space Y into X. Thenf
is continuousif
andonly if
pri of is
continuousfor
eachi.
Proof. Assume that pri o
f
is continuous for each i. Let y E Y and T C Ysatisfy apart ( f (y), f (T)) . By Proposition 4,
there existVI
CX, and V2
CX2
such thatf (y)
E-Yl
x-V2
andSetting
146
we have T C
Tl
UT2. Moreover,
for eachi,
pri of (Ti) C Vi
and thereforeOur
continuity hypothesis
now ensures that y E-Tl
n-T2
c -T.The converse follows
easily
from Lemma 5.The local
decomposability
of the spacesX,
andX2
is usedonly
oncein the
foregoing proof: namely,
at the invocation ofProposition
4 in orderto
produce
the setsV1
andV2.
Thissuggests
that instead ofdefining
theproduct point-set apartness
as wedid,
wemight
have defined itby taking apart (x, A)
to mean that condition(*)
ofProposition
4 holds.However,
thestatus of axiom A4 for this notion of
’apart’
remains undecided: we know of neither aproof
that A4 holds nor a Brouwerianexample indicating
that A4is not
constructively
derivable.3 Set-set apartness
We now introduce a notion of
apartness
between subsets. We first assumethat there is a set-set
pre-apartness
relation m betweenpairs
of subsetsof
X,
such that thefollowing
axioms hold.For the purposes of this paper, we then call X a
pre-apartness
space, or, ifclarity demands,
a set-setpre-apartness
space. We then defineIf,
in addition toB1-B4,
the axiom147
holds,
then we call m a(set-set) apartness
onX,
and X a(set-set)
apartness
space.The
morphisms
in thecategory
of set-set(pre)apartness
spaces are thosefunctions
f :
X-> Y that arestrongly continuous,
in the sense thatGiven a
apartness
spaceX,
we defineand
to obtain the
inequality
and thepoint-set apartness
relation associated with thegiven
set-set one. Thepoint-set
relation pa satisfies axiomsA1-A4,
where the
apartness complement
is defined as at(3).
If the set-set relationalso satisfies
B5,
then thepoint
setapartness
islocally decomposable
and afortiori
satisfies A5.An
example
of a set-setapartness
space is affordedby
a uniform space(X, u ) ,
on which theinequality
is definedby
It is shown in
[7]
that the relation m defined forsubsets S,
T of Xby
is a set-set
apartness relation, Denoting by apartu
theapartness
relation associated with thetopology
Tu induced on Xby U,
we can show that x m S if andonly
ifapartu (x, S’) .
The definition of the
product
of two set-setpre-apartnesses
is much morecomplicated
than that forpoint-set apartnesses.
If X =Xl
xX2,
whereX,
and
X2
are set-setpre-apartness
spaces,then, taking
thecontrapositive
ofthe definition used on page 23 of
[6],
we define theproduct apartness
on X as follows. Two subsetsA,
B of X areapart,
and we write A mB,
ifthere exist
finitely
many subsetsAi (1
im)
andBj (1 j n)
of Xsuch that
148
and
Proposition
7 LetXl, X2 be pre-apartness
spaces, and X their Cartesianproduct, define
the relationapart
as at(1),
and let > denote theproduct pre-apartness
relation on X. Then{x}
m S entailsapart (x, S) . Moreover, if
bothX,
andX2
areapartness
spaces, thenapart (x, S)
entails{x}
> S.Proof. Assume first
that {(X1,X2)} >]
,S’. Then there exist subsetsBj (1 j n)
of X such that ,S CBI U...UBn,
and foreach j
either x, mpr1Bj
or z2 m
pr2Bj . Renumbering
the setsBj
if necessary, we may assume that there exists v n such that x 1 mpr 1 Bj
for1 j v,
and X2 mpr2Bj
forv + 1 j n .
WriteThen x, E -U and X2 E -V. Given
x’1
E-U, x2
E-V,
and(E,n)
ES,
choose j
such that(E, n)
EBj .
If1 j
v, thenx’ # E;
if v +1 j
n, thenx’2 # n.
Hence(x’, x’2) # (E, n).
We now see that-in other
words, apart (x, S) .
Now assume that
X1, X2
areactually
set-setapartness
spaces and thatapart (x, S’) .
So there existUi C Xi
withWe can find
sets v C Xi
such thatSet
For
each E
in S wehave Çi E Vi
for some i(for
ifgj
E-Ui
for bothi,
thenE E -U1 x - U2
E NS,
acontradiction);
whence S CBl
UB2. Moreover, by
our choice of
V ,
wehave xi
E-V ;
whencefxil = pri(A1)
mV - pri(Bi).
Thus
{x} > S.O
149
Having
shown that our definition ofproduct
set-setapartness
iscompat-
ible with the earlier definition ofproduct point-set apartness,
we still need toverify that,
in thelight
of axiomsB1-B5,
we haveactually
defined anapartness
on the Cartesianproduct
X =Xl
xX2
of the setsunderlying
the
apartness
spacesXl
andX2.
Onceagain,
wekeep
track of theneed,
orotherwise,
for axiom B5.Proposition
8 Let X =X,
xX2
be theproduct of
two set-setpre-apartness
spaces. Then the relation m,
defined
as abovefor X, satisfies
axiomsB1-B4.
If X1, X2
areapartness
spaces, then m alsosatisfies
axiom B5.Proof. We
verify
each of the axioms in turn.B2 Let
A,
B C X =X,
XX2
and A > B. Choose the setsAi, Bj
as in thedefinition of m on X.
Supposing
that(x, y)
E A nB,
choosei, j
suchthat x
E A2
and y EBj.
Then(x, y)
EAin Bj,
so x E prl(Ai) n prl (Bj)
and y E pr2
(Ai) n
pr2(Bj) ,
which contradicts theproperties
ofAi
andBj .
B3 Let
R, S, T
be subsets of X =X1
xX2 .
It is easy to prove that if R m(S
UT),
then R m S and R > T.Suppose
then that R m S and R m T. There exist subsetsR1, ... , Ra
andR’1, ... , R)
ofR,
subsets81, ... Sm
ofS,
and subsetsTl, ... , Tn
of T such thatfor 1 i a and
1 j
m there exists k such thatprk (Ri)
mprk
(S,) ,
and for 1 iB
and1 j
n there exists k such thatprk(Ri) >
prk(tri).
LetThen
150
Also,
ifI x
m,then, choosing k
such that , we seethat
so prk
(Pi,j)
> prk(5t). Likewise, if l
n, then there exists k such that prk(Pi,j)
> prk(Ti).
Sinceit follows that R m
(S
UT) .
B4 This is trivial.
This
completes
theproof
that X is apre-apartness
space. Now suppose thatX1, X2
areactually apartness
spaces.Then, regarded
as apoint-set apartness
space, eachXi
islocally decomposable; whence, by Proposition 2,
the
product point-set apartness
space X islocally decomposable.
It followsfrom
Proposition
7 that theproduct point-set apartness
on X isprecisely
the
point-set
relation inducedby
theproduct
set-setapartness
on X. HenceX satisfies B5.
It is almost immediate from the definition of the
product
set-set pre-apartness
structure on X =X l
xX2
that theprojection
maps pri : X ->Xi
are
strongly
continuous. We end the paperby showing
that theproduct
pre-apartness
structure has the characteristicproperty
of acategorical product.
Proposition
9 Let X =X,
xX2
be theproduct of
two set-setpre-apartness
spaces, and
f
amapping of
a set-setpre-apartness
space Y into X. Thenf
is
strongly
continuousif
andonly if pr2
of
isstrongly
continuousfor
each i.Proof. Assume that pri o
f
isstrongly
continuous for each i. Let,S’, T
besubsets of Y such that
f (S)
mf (T),
and choosefinitely
many subsetsAi
(1
im) and Bj (1 j n)
of X such that, and
O for all
i, j either
]-151-
For all
i, j
write,S’i
=f-1 (Ai)
andTj
=f -1 (Bj).
Then S’ cS1
U ... USm
andT C
T1 U...UTn. Moreover,
for alli, j
we have eitherpr1 o f (Si)
papr 1 o f (Tj)
orpr2of(Si) > pr2 o f (Tj); whence Sj
paTj, by
ourstrong continuity hypothesis.
It follows that S m T and hence that
f
isstrongly
continuous.The converse
readily
follows from thecontinuity
of theprojection
maps.Thus we have shown that under reasonable
conditions,
such as local de-composability
in thepoint-set
case, both thepoint-set product apartness
and the set-setproduct (pre)apartness
arecategorical.
This is another indi- cation that the notion ofapartness
may have some constructive merit.In earlier papers such as
[3],
we haverequired
thatpoint-set apartness
spaces be nontrivial. This
requirement
would mean that in the context ofproduct
apartnessstructures,
we would not havenullary products,
which arethe terminal
objects
in thecategory.
Much has been done in the four years since the
theory
ofapartness
spaceswas first broached as a
possible
way ofapproaching topology constructively.
In
particular,
connections betweenpoint-set apartness
andtopology,
and be-tween set-set
apartness
anduniformity,
have beenexplored
in somedepth;
see
[3, 4 , 7, 8, 9] . Among
theinteresting phenomena
observable with intuition- isticlogic
is that if certain natural set-setapartness
structures are induced(as they
are under classicallogic) by
uniform structures, then the weak law of excludedmiddle,
-p V
P7
holds;
see[8] (Proposition
4.2 andCorollary 4.3). Thus,
in a certain sense, theconstructive
theory
ofapartness
spaces islarger
than thetheory
of uniformspaces.
Of the
major problems
that remain in the foundations ofapartness-space theory,
the mostsignificant
and resistible to constructive attack is that ofcompactness.
Since the Heine-Borelproperty
fails to hold for the interval[0,1]
in constructive mathematicsplus
Church’s thesis([2], Chapter 3),
andsince the
sequential compactness
of[0, 1]
isessentially nonconstructive,
formetric and uniform spaces we define
compact
to meantotally
bounded andcomplete. Lifting
such ideas from the uniform- to thegeneral apartness-
space context appears to be a
highly
nontrivialexercise,
for which we haveas
yet only partial
solutions(see [5]).
152
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2003.Authors’ Addresses:
Bridges
andVîta: Department
of Mathematics &Statistics, University
ofCanterbury,
Private
Bag 4800, Christchurch,
New Zealand.email:
douglas . bridges@canterbury .
ac . nz153
Ishihara: School of Information
Science,
Japan
Advanced Institute of Science andTechnology, Tatsunokuchi,
Ishikawa923-1292, Japan.
email:
ishihara0jaist.ac.jp
Schuster: Mathematisches
Institut,
Ludwig-Maximilians-Universitat Munchen,
TheresienstraBe
39,
80333Munchen, Germany.
email: pet er . schuster@mathematik.uni-muenchen.de