PROBABLISTIC CONVERGENCE SPACES

VOL. 20 NO. 4 (1997) 637-646

637

PROBABLISTIC CONVERGENCE SPACES

AND REGULARITY

P.BROCKand D.C.KENT

Department

ofPureand AppliedMathematics WashingtonStateUniversity

Pullman, WA99164-3113, U.S.A.

(Received February 5, 1996 and in revised form May 20, 1996)

ABSTRACT. The usual definitionofregularity for convergence spacescan becharacterizedby ^a diagonalaxiomRdueto Cook andFischer. Thegeneralizationof

R

^tothe realm of probabilistic convergence spaces depends on a t-norm

T,

and the resulting axiom

Rr

^defines "T-regularity", which is the primary focus of this paper. Wegive several characterizations ofT-regularity,both in generaland forspecificchoicesof

T,

andinvestigatesomeof its basic properties.

KEY

WORDS

AND

PHRASES. convergencespace, probabilistic convergence space, t-norm, T-regularspace,T-approachablespace,category

1991 AMS SUBJECT CLASSIFICATION CODES: 54 A 05, 54 A _20, 54

D

10, ^5,1

E

70

INTRODUCTION.

Probabilisticconvergencespaceswereintroduced in 1989byL. Florescu

[4]

^intermsofnets,and werelater reformulated usingfilters in

[8].

Probabilisticconvergencespacesare anaturalextension ofprobabilisticmetricspaces

[9];

the fundamentalideais toassigna numericalprobabilitytothe convergence ofagivenfilter toagiven point. Fromacategorical perspective, the category

PCS

of probabilistic convergence spacesis cartesianclosed and hereditary

(i.e.,

^aquasitopos).

In [3],

^Cook and Fischerinvestigated two diagonal conditions for convergence spaces, ^wecall them

F

and

R,

whichareinanatural way dual to each other. A convergencespaceistopological iff it satisfies

F,

and regular iffit satisfies R. In

[8],

^{F was} generalized relative to an arbitrary t-norm

T

^toanaxiom

Fr

for probabilistic convergencespaces, and itwasshownin

[2]

^{that thefull} subcategoryofPCS determinedbythe left-continuousprobabilisticlimit spaces satisfying

Fr

^for

astrictt-norm

T

isisomorphictoR.

Lowen’s

category

AP

ofapproachspaces

[7].

^{Thus wedefine}

probabilistic convergencespacessatisfying

Fr

^tobe T-approachable.

Our goalin this paperistostudythedualaxiom

Rr

^{in thecategoryPCS.} Probabilisticconver- gence spaces satisfying

Rr

^{aredefinedto be}T-regular. IfT and

T’

^{aret-normssuch that}

T<T’,

^then

T’-regularity

implies T-regularity. Indeed,thelargest t-norm, denoted by 7

,

^inducesthe strongest type of "T-regularity," which is equivalent to "componentwiseregularity". Likewisethe smallest t-norm,

,

^{induceswhatwe}naturally call "weakregularity".

For an arbitrary t-norm

T,

we give several "traditional" characterizations of T-regularityin

PCS

andsome of itssubcategories,alongwithexamplesto show howT-regularity dependsonthe choiceofT. Wealsoshow that T-regularity^ispreservedunder theformationofinitialstructures, thereby demonstrating that the category

RrPCS

^ofT-regular probabilisticconvergence spaces ^is

638 P. BROCK AND D. C. KENT

bireflectivelyembedded in PCS. It should also be noted that the category RCONV ofregular convergencespacesisbicoreflectivelyembedded in

RrPCS

^foranarbitraryt-normT.

Inthe lastsection,weshow that incertainsubcategoriesof

PCS,

theaxiom

Fr

^and

Rr

^canbe

stated entirely in terms ofultrafilters,^{whichin some}situationsresults inanappreciable simplifica- tion.

1. CONVERGENCE SPACES.

Let

X

be aset,

F(X)

the set of all filters on

X, U(X)

^{the set ofall}ultrafilters on

X,

and 2x thepowersetof

X. For

eachzE

X, :

denotes the fixedultrafiltergeneratedby z.

Definition 1.1. A function q

F(X)

^2xiscalledaconvergence structureon

X

ifitsatisfies thefollowingaxioms.

(C,)

^zE

q(:),

forallz q

X;

(c) c_ q(y) q();

(Ca)

^z

(q()

6q(

).

If_{q is}aconvergence structureon

X,

the pair

(X,

q)^iscalledaconvergencespace,and "x

q()"

willusually be written

" ^{z" (Y}

q-converges to

). A

function

f’(X,

q)

(Y,p)

^between

convergencespaces is continuous if

f() f(x)

^{whenever x. If}p and _q are convergence structureson

X

and

f"

^(X,q)

(X,p)

^iscontinuous,where

f

^{istheidentitymapon}

^X,

^thenwe

write p q(piscoarserthan q,orq is

finer

^thanp).

Foraconvergencespace

(X,

q),consider thefollowingadditionalaxioms.

(C4)

^{zand z z.}

(Cs)

^If

F(X)

^and

^Y

^{z, forall}

Y

^q

U(X)

^suchthat

9 ,

^then

G &

z.

(C6) Pq(z)

z,for allz

X,

whereFq(z)^isthe q-neighborhood

filter

^{atz, definedby}

A

convergence structureqon

X

satisfying

(C4)

(respectively,

(C5), (C6))

^{iscalledalimitstctu} (respectively,pseudotopolo9y, pretopolo99) and

(X, q)

^iscalled a limit space(respectively,pseudo- topologicalspace, pretopological

space).

With every convergencespace

(X, q),

^{there is an} sociated closure operator clq 2x 2x definedbyclqA

{z ^X"

^{zsuch thatAq}

Y},

for allA

G

^X.

(X,

q)^issaidto beregular if z impli

clq

z, whereclqisthefiltergenerated by

{clef "’F _}. A

convergence space is saidtobetopologicalifq-convergencecoincidwiththatrelative tosometopoloon

X;

in thisceit iscustomarytoidentifythat topologywith q.

Itisaninteresting

(and

apparentlynot

well-known)

factthat theconvergenceproperties

"regu-

lar" and "topological" areinavery naturalsensedualtoeachother,^sincetheycanbe characterized bymeansof dual axioms, whichwecall

F

and

R,

due toC.H. Cook andH.R. Fischer

[3].

^t

^X

and

J

be non-empty sets, q

F(J),

^anda"3

F(X).

^Wedefine

tiscalledthe "compression operator for

"

^relativetotr." Notethat if.7"

U(J),

^and

tr(y)

E

U(X)

for ally5d,then

a"

q

U(X).

^{Wecannowdefinethe axioms}

F

andR.

F: Let

J

beanon-empty set,

k"

^d

^X,

^andletr-d-,

F(X)

^{have thepropertythat}

tr(y) & _b(y),

for_{all y q} d. If

" ^F(J)

^issuchthat

ff(2") -%

x, thenr."

-,q

_z.

R: Let Jbeanon-emptyset,

"

^J

^X,

^{and lettr"J}

^F(X)

^{have thepropertythattr(y)}

^-,q

^(y),

for all yEJ. If

Y

E

F(J)

^issuchthat #.aY x, then

(Y)

z.

The next propositionsummarizespreviously mentioned results pertaining to theseaxioms. The first assertionisprovedin

[8],

the second in

[1]

^and

[31.

Proposition 1.2. Let

(X,

q) beaconvergencespace.

(1) (X,q)

istopologicalifand onlyif it satisfiesF

(2) (X, q)

is

regul.ar

^ifandonlyif it satisfies

R

Let F andR denote theaxiomsobtainedwhen

"F(X)"

^is replaced by

"U(X)"

ⁱⁿFand

R,

respectively. Obviously,

F = ^F"

^and

^R =

^R

.

The next proposition isprovedin

[5].

Proposition 1.3. Forconvergencespaces,

F == ^F"

^and

^R == ^R .

Let CONVdenote thecategory ofconvergencespaces and continuous maps. Let RCONVbe the fullsubcategory ofCONV determinedby theregularobjects and TOP the fullsubcategory

’of

CONVdetermined by the topological objects. Itis well known that both

RCONV

and TOP

arebireflectivesubcategoriesof

CONV,

since the properties "regular" and "topological" ^areboth preservedunder formation of initial structures.

2. PROBABILISTIC CONVERGENCE SPACES.

This section is mainlya review ofrelevant definitions and theorems from

[2]

^and

[8].

Let

I

denote the closedunitinterval

[0,1].

Definition 2.1. Aprobabilisticconvergence structure qon Xis afunction q

F(X)

I 2x which satisfies:

(PCS1)

Foreach p q

I, q(,p)= %(),

where each % is^aconvergence structureon

X;

(PCS2)

^Whenp 0,_%isthe indiscretetopology;

(PCSz) If/ _<

t/,then q.

_<

q..

If qisaprobabilisticconvergencestructureon

X,

the pair

(X, q)

^{is calleda}probabilistic ^con- vergence space. We will usually write q

(q,),

^{where it} isunderstood that _ft ranges through

I;

the

q,,’s

arecalled the "component convergence structures." If q

(q,)

where each qu ^{is a} limit structure (respectively, pseudotopology, pretopology, topology), then

(X,

q)is calleda probabilis- ticlimit space(respectively,probabilistic pseudotopologicalspace, probabilistic pretopologicalspace, probabilistic topologicalspace).

If

(X,

q)isaprobabilisticconvergence space, q,

" ^F(X)

^andx

^X,

^thenp

sup{t/ I"

z}

isinterpretedastheprobabilitythat

.

q-convergestoz.

A

probabilisticconvergencespace

(X, q)

^is

left-continuous

if,foreachp E

(0, II,

qu

sup{q

u

< p},

and constant if, for eachg

(0,1],

qu ql.

If

(X,

q),

(Y, p)

areprobabilistic convergencespaces and

f: X

^Yisamap,then

f: ^(X,

^q)--,

640 P. BROCK AND D. C. KENT

(Y,

p) is said to be continuousif

f" (X,

qu)

(Y, Pu)

is continuous,for

each/

^E I. We^denote byPCSthecategorywhose objectsareprobabilistic convergence spaces and whose morphismsare continuousmaps. SomefullsubcategoriesofPCS whichareofinterestarethefollowing:

PPSS (objectsareprobabilisticpseudotopologicalspaces) PPRS (objectsareprobabilisticpretopologicalspaces) PTS (objectsareprobabilistic topologicalspaces)

Furthermore,thefullsubcategory ofPCSdeterminedbythe constant objectsis isomorphicto CONV. Wenotethat

CONV, PTS, PPRS,

andPPSS arebireflectively embeddedin

PCS;

CONV isalso bicoreflectively embeddedin

PCS.

Thenotionof

"t-norm,"

whichisvital in thestudy of probabilisticmetric spaces

(see [9]),

^also

playsanimportant role inthe study of probabilistic convergencespaces. Weshallsummarizesome factsabout t-norms whicharerelevant to this paper; for further informationthereaderisreferred to

[2]

^or

[9].

At-normisabinary operatorT

12

Iwhich isassociative, commutative, increasingin each variable,and satisfies T(p,

1)

^{p, for all}# E I. Let Tbe the setof allt-norms, with pointwise partial order.

T

contains a largest member

,

^{defined by}

(p,v) min{#,v},

^{and a smallest}

ember ,

^definedby

/s, ifv=

J’(/, v)

v, ifp 0, otherwise.

A

t-norm

T T

^issaidtobestrictif there isasurjective, strictly decreasingmapS 1

[0, oo]

such thatT(p,

v) S-I(S(/) + S(v)).

^Neither

"

^{nor7 isstrict; anexampleofa}strict t-norm is

T(/z, v)

=/zv,where

S(/z)

log/.

Let

(X,

q)^beaprobabilisticconvergencespace, and letT T. Wedefine twoaxiomsfor

(X, q)

relative to

T

whicharederived inanobvious way from the axioms

F

andRof Section 1.

FT

^Let

^a(y)

^p,

_-

^v

^(y),

^{I. Let Jforeachbe anyy J.non-emptyIf}

^.T"

^E

^F(J)

^set,and

-

^J

^{" L}

^{Xx, thenanda"}

^a"

^{J q’(,,)}

^F(X)

^{z.besuch that}

RT

^{Letp,v I. LetJbe anynon-empty set,}

"

^J

^X

^anda"J

^F(X)

be such that a(y) (y),for each y J. If

F(J)

^and

a ^2.

x,then

" ^q,rL;

^)x.

ForafixedT

T,

the fullsubcategoryofPCSwhoseobjectssatisfy

FT

^{is denoted by}

FTPCS.

The next threepropositionssummarizesomealreadyknownresults.

Proposition ^2.2.

[8]

(1) FTPCS

^isabireflectivesubcategoryof

PCS,

forevery

T

E

T;

(2) PTSC_FTPCSC_PPRS,

forevery

T T;

(3)

^If

^T ,

^then

FTPCS=PTS.

In [7],

^{R. Lowenintroduced the}category

AP

ofapproachspaces which contains the categories TOPand

MET (metric

spaces and non-expansive

maps)

asfull subcategories.

In [6], R.

Lowenand E. Lowenembedded

AP

inaquasitopos CAPofconvergence approachspaces.

As

aconsequence of

the next proposition, bothCAPandAP arebireflectivelyembedded inPCS.

Proposition

2.3[2].

Let

T

beany strictt-norm.

(1) AP

isisomorphictothe fullsubcategoryof

FTPCS

whose objectsareleft-continuous limit spaces.

(2)

CAPisisomorphictothefullsubcategoryofPCSwhose objectsareleft-continuous limit spaces.

In

viewof thefirst assertionofProposition2.3,^wedefineobjectsin

FTPCS

^tobeT-approachable.

In

viewof Proposition 2.2

(3),

^the

-approachable

probabilistic convergence spaces areprecisely the probabilistictopologicalspaces, which yieldsadirectgeneralizationof Proposition 1.2

(1).

Proposition

2.4[8].

^Foraprobabilisticpretopologicalspace

(X,

q),andat-norm

T,

thefollowing statementsareequivalent.

(1) (X,

q)isT-approachable.

(2)

Forarbitrary/,v5

I

and for eachV^q

i)qr. ^(z),

there existsWE

I)q(z)

^suchthat,for each

W,

V

V().

(3)

^Forarbitrary/,v EIandAC_

X, cl(clq(A))

^C_

clr.)(A).

’3.

T-REGULARITY.

Aprobabilisticconvergencespace

(X,

q)isdefined tobeT-regularifitsatisfiesthe axiom TheT-regularobjectsin PCSdeterminethe fullsubcategory

RrPCS.

Theorem 3.1. Let

(X,

q)

[PCSI,T

T. Then

(X,

q) isT-regular iff,for all

,

^v

e I,

2"

implies

cI2-

^x.

Proof. Assumethat

(X, q)

^isT-regular,and let2- x. Forarbitrary/z

I,

let

Y {(7,,) 7 u(x), x,; u}.

Definea-J

F(X)

by a({,y)

,

^and

p

J

-- ^X

^by

^h({,y)

^{y. Then}

^a(z) - ^b(z)

^holds

forallz E J. Foreach

F

2-,let

SF {({,y)

^J"

F

^q

},

andletS be the filteron Jwith base

{5’F" F 2-}.

Oneeasilyverifies that2- C_

a5’,

^andhence(S

’x. By RT,

itfollows that

k(S)

Conversely,

clq2-

let

- J,

^x,a,^as

p,

^desired.and2- E

F(J)

^beasin the statementof

RT,

andassumethattr2- z.

Since

tr(y)

b(y),forall y

J,

one may confirm that

cl.tr2-

^C_

2-.

^But

cltr2- ^;)

^zby

q’(...,)

hypothesis, andconsequently

k2-

^{x. Thus}

(X, q)

satisfies

RT.

Corollary3.2. Let

(X,

q) beaprobabilistic convergencespace.

(1)

^If

T,T’

Tand

T’ < T,

^{then if}(X,q) ^isT-regular,it isalso

T’-regular.

(2)

If

(X,

q)^isT-regularforsomeT Tand2-

25

_z,then

cl.

^z holds for all g I.

In

particular, if

(X, q)

isT-regular,thenql isaregularconvergence structure.

(3) (X, q)

is

-regular

^iff

(X, q)

iscomponentwiseregular

(i.e.,

q,isaregular convergencestructure for all t E

I).

(4) (X,

q)is

-regular

iffboth of thefollowinghold:

(i) "

^z=,,

^clq,,2- -

^x,forallt

^I

642 P. BROCK AND D. C. KENT

(ii) .

^z

⁼

^clq,"

-

^{z, forallftE}

^I

Proof. Allofthese results followdirectlyfrom Theorem 3.1. Inparticular,the first assertion in

(2)

followsby taking^v 1, and the second by letting/ ^r, 1. Toprove

(3),

let/ ^uand note that

’(,,

g)

, _^ ,.

In

thesubcategoriesPPRS andPTSof

PCS,

the characterization ofT-regularity givenin Theorem3.1 canbereformulatedasfollows.

Corollary3.3.

(1)

(X,q)G

[PPRS[

^is T-regular iff,for all p,uGIand

(2) (x,

q)

IPTSl

^{isT-regulariff thefollowing}holds for allp,v I: Ifz

X

and AC_X is a

qrt,,)-closed

^setnot containing z, then there^isaq,-open neighborhood

U

ofz anda q-openneighborhood VofAsuch that UNV }.

A probabilistic convergence space is said to be strongly regularif it is

5b-regular

and weakly regularifitis

-regular.

Itfollowsby Corollary3.2

(4)

^{that every}

(X, q)

E

IPCSI

^{such that ql is}

discrete isweaklyregular,demonstratingawidegapbetweenweak andstrong regularity. Notethat

"strong

regularity" accordingtoourdefinition coincides with "regularity" asdefined in

[8].

Example 3.4. LetTbe the t-norm defined byT(p,

t,)

pv,letX beanyinfiniteset,and let q be theprobabilistic convergencestructure definedby

discretetopology, p

(1/2,1]

q. cofinitetopology, p

e (, ]

indiscretetopology, p

e [0, ]

Weobtaina T-regular space

(X,

q) which is not strongly regular. Howeverifwemodify qonly slightlytoobtain p definedby

discretetopology, /q

(, 1]

p. cofinitetopology,

u

^E

[1/4, ]

indiscretetopology, _t

[0, 1/4),

theresulting probabilisticconvergence space

(X,

p)isweaklyregularbut notT-regular.

Example3.5. Weborrowfrom

[8]

^anexample of^aprobabilisticconvergence space whichisstrongly regularbut not T-approachablefor any t-normT. LetA denoteLebesguemeasureandrthe usual topologyon

I [0,1].

^Let

X

be thesetof allreal-valued, Lebesgue-measurablefunctionson

I,

the convergencestructure % on

X

isdefinedasfollows:

- ^f

^{iff thereisAC_Isuch that}

^{A(A) <}

^1-p

and

Y-(v)

^2,

f(v),

^{for all}v

I-

A.

One

easilyverifiesthatq

(q,)

isaprobabilisticconvergence structureonX.

Forarbitrary_ft^q

I,

let

f.

^{Then thereisAC_}

^I

^{such that}

(A) <

-/and

’(v)

^2,

f(v),

for allv

I-

A.

By

regularity of’,

cl.T’(v) -h f(v)

^{for allv}E

I- A,

and

(clq,,.$’)(v) >_ cl,(.T’(v))

implies that

(clq.)(v) ^-h f(v)

^{for allv}

^I-

^{A. Thus}

clqff" - ^f,

establishing that q^isstrongly

regular. The fact that

(X,

q) ^isnotT-approachablefor any

T T

isprovedin Example 3.13,

[8].

Example3.6. Let

X

betheset described in Example3.5 and let

Y

^bethe set obtainedfrom

X

by identifyingfunctions whichareequalalmost everywhere. Letp betheprobabilisticconvergence structureonYdefinedasfollows: _{for each t}

(0,1],

2-

- ^f

^{inYiff,for eacha>0ande</there}

isf 2-suchthat, for eachg

F, {v

I’lg(v

f(v)l < a} ^>_

^{e. For}t 1, px isconvergence in probability. Let

T’

be the t-norm defined for g,v Iby

T’(l,v) max{t +

^v-

1,0}.

^Itis

shown in Example 3.14,

[81,

^that

(Y,

p) is

T’-approachable.

We shallnow showthat

(Y,

p) is also

T’-regular.

Leta

>

0,

,

^v

I,

and2-

- ^f.

^Let

^{T’(g, v)}

^{t/. If O, clpv2-}

beanumber such that 0

<

e

<

/. Since/=/

+

^{v- 1,wecanchooseel}

^<

t ande2

<

vsuch that

ex

+

^e2

^>

^{e. Then}

" ^f

implies thereisFE

"

^{such that}

^{{ l’lh()}

Ifg clq.F,there is

-

^{g withF}

^G;

thus there is G such that

,{ l’lk()- g()] <

} ^>_

^{e2for allk G. Let}

^h’

^{Gf3F. Then}

{

l’ig()-f()]

< a} >_ A{ I" [g()-h’()l+

Ih’()-f(OI ^{< ,} >} a [{ ^{e z"}

^ig()-

h’()l < } ⁿ { Z-Ih’() f()l < }] ^>

Therefore

cl,,..T"

f,which establishesthat

(Y,

p)is

T’-regular.

Inparticular,

(Y,

p)is weakly regular. However

(Y,

p)fails to be T-regularfor

T(U, u) u.

Indeed,let2-

],

where

f

Xt ^isthe characteristic function for I. Let

where g X[o,]. ^{Then 2-}

X[,a],

^and

>_ c12-.

^But

^G

^{fails to}

^p-converge

^{to X[1/2,l and it}

follows that

(Y, p)

^{is not} T-regular.

Theorem 3.7. Forafixedt-norm

T,

let

{(Y,,p)

t

A}

^{beacollectionofobjects in}

RTPCS.

Let

X

beasetandf,, X

Y

^afunction,forallcEA. Ifq is the initialstructureon

X

relative tothe families

{(Y,,, p)’a A}

^and

{f,,

^cr

A},

^then

(X,

q)isT-regular.

Proof. Asisnoted in

[8],

^{for any v}

I," 2

xiff

fo(.T’) - ^fo(x),

^{for alla A. Thusfor/,v}

^e

I,

2- ^:r

impliescl,(f,,,(2-)) ]’o(x)in (Yo,p(,}),

^{for alla A. Since}

o ^{(X,q) (Yo,p’)is}

continuousforallp

I, c/,:(f(2-))

^C_

fo,(clq,(.T’)),

and hence

fo(c/,,(2-))

^--,

fo(x)in (Yo,P(t,,))

holds for eacha

A,

since every

(Y=,

p")^is T-regular. Consequently,

clq,,(2-)

^{---,x in}

(X, qT(,,,,)),

establishingthat

(X,

q)^isT-regular.

Corollary3.8. T-regularityispreserved undersubspacesand arbitraryproducts in

PCS.

Further- more,

RTPCS

^is bireflectively embeddedin PCSforany

T

T.

Corollary 3.9. CONV and RCONVarebicoreflectivelyembedded in PCS and

RrPCS,

respec- tively,for anyT T.

Proof.

In

bothcasesthe bicoreflector maps

(X, q)

to

(X, q)

and preserves the underlyingfunction.

Note

that q is regularwheneverq isT-regularbyCorollary 3.2

(2).

Wehaveseen inExample 3.3 that T-regularitydoesnot generally imply "componentwise reg- ularity," and the question naturally ^arises whether there is some weaker property which every componentstructure ofaT-regularprobabilistic convergence structure must satisfy. Indeed,there is such apropertywhichwedefine as follows: aconvergencespace

(X, q)

is symmetricif,for all x,y

. ^X,

^{ximplies y. Notethateveryregular}convergence structure is symmetric.

644 P. BROCK AND D. C. KENT

Proposition3.10. Aprobabilistic convergence space

(X,

q)which isT-regularfor any

T

E

T

has theproperty that eachcomponent convergencespace

(X,

qu)issymmetric.

Proof. Let_{t E} Iand let x. Let J

X,

let bethe identity mapon

X,

andleta X

F(X)

be defined by

a(x)

and

a(z) ,

^{for z}

#

^{x. Then tcak}

,

^and by

R-, :

y, since /

T(/z, 1).

We close this section with a simplecharacterizationof those probabilistic convergence spaces whicharesimultaneouslyT-approachableandT-regular.

Proposition 3.11.

(X, q) IFTPCS[

⁽³

RTPCSI

^iffthe followingconditionsaresatisfied:

(1) clq(Vq(z))

^zin

(X, qT(t,,t,)),

^for

all/,v e I

andx

e X.

(2) clq,,(cl,,,(A))

^C

c/qr(.)(A),

^forall/z,vq

^I

^and

^A

^_CX.

Theproofisaneasy consequence of Proposition 2.4 andTheorem3.1.

4.

ALTERNATE

FORMULATIONS OF

THE

AXIOMS.

The results of thepreceding section pertainingto

RT,

combined with those of

[2]

^and

[8]

^in-

volving

FT,

demonstratethe usefulness of these axioms in the study ofprobabilistic convergence spaces. Furthermore

RT,

whentranslated under theisomorphismmentioned in Proposition2.3, has important ramifications inthe study ofapproachspaces,convergence approachspaces,andrelated categorieswhichweshalldiscusselsewhere.

Inworkingwith axiomsordefinitionsbasedonfilters,^itisuseful toknow when =filter" canbe replaced by "ultrafilter" withnoresultingloss ofgenerality. This istrue, for instance,in defining

"Hausdorff" inthe settingofconvergencespaces;anadditional illustration isfoundin Proposition 1.3.

In

thisconcludingsection weshow that Proposition 1.3canbegeneralizedtotheaxioms

FT

and

RT

for arbitrary probabilistic convergence spaces. Furthermore, by restricting

FT

^and

RT

^to

thecategory PPSSofprobabilisticpseudotopological spaces,theseaxiomscanbe given equivalent formulationsbased entirelyonultrafilterconvergence.

Foraprobabilistic convergencespace

(X,

q)anda

-norm T,

let

F ^be.the

^axiomobtained when

"F(X)"

^isreplaced by

=U(X)"

^{in theaxiom}

FT.

Furthermore,let

F."

^betheaxiomobtained when

"F(J)"

^is replaced by

=U(J)"

^{inthe axiom}

F. ^In

^{exactly thesameway,wederive}

^Rr

r from

Rr

and

1"

^from

^Rr

r.

It

isobviousthat

FT => F, => F,"

^and

RT ^=>

^R:r

= ^Per’.

Theorem4.1. For

(X, q)

E

]PCS]

^and

T e T, FT ==> F.

^and

RT 1.

Proof. Assume

F. ^By

Proposition2.4, itsuffices to show that

(X,

q)^is aprobabilisticpretopo- logicalspace which satisfies:

clq,,(cl,,(A))

^C_

clqrc,,.,o(A),

^{forall/z,v q}

^I

^and

^A

^{_CX. Theproofof}

Proposition 3.5,

[8],

^{estabishesthat if}

(X,

q) satisfies

F.,

^then

(X, q)

^is pretopological. Toprove the assertion about closures, letx

cl,(clq,,(A));

^{then there is an} ultrafilter 7"/

!

x such that

cl,,(A)

E 7"l. Let

J

Xand

(y)

y,forall y X. Ify

e cl,,(A)

^{there isan}ultrafilter y such that

A e G.

^Define

^{a(y) G,}

^fory

cl,.(A)

^and

^a(y) ,

^{if y}

. ^cl,,(A).

If follows that

A tca’H. By FT, a7-/"’)

x, and thereforex

clqro,.,,)(A ^).

Next,

assume

l:t..

^{To establish l:tr, it} suffices to prove the characterization of

RT

^{given in}

Theorem 3.1. But inthe first half of the proof of Theorem3.1,we observe that

a(z)

E

U(X)

for everyzE J. Thus thissameproofremainsvalidwhen

Rr

^{isreplacedbyR:}r.

Lemma4.2. Let jr, a, and

" ^F(J)

^{be asin}the statement of

F..

^If/4

^U(X)

^and

> na.T’,

then thereexistsa

G

E

U(J), G > -

^{such that}

^nag

Proof. LetUC

U(X)

be such thatU

>

na.T’. For AC/4,define

HA

{yC J" AC

a(y)}.

^Observe

that

HA

^N

^{F 1}

^forall

^F

⁶

"

^{and all}

^A

^6/4;otherwise, for some

F

.7:’,

X\A a(y)

for all

y6 F. But thisimplies

X\A naY,

which is acontradiction. Let

"H

be the filtergeneratedby

{HA

^A

6/A},

^{and let}

G

^beany ultrafilteron

J

with the property that

G >

"

^{V 7"/. Itiseasyto}

verify that

aG

^{=/4. |}

Theorem4.3.

For (X, q) IPPSSI

^and

^T

^E

^{T, FT} == ^F’.

Proof. Assume

F,’.

^Let

J,

a,

,

^and

_"

6

U(J)

^{be as in thestatementof}

F.,

^{and assumethat}

.T" ²⁵

^{x. Let} H

U(X)

be such thatH > a’. By Lemma 4.2, thereexistsa

G U(J),

^such

that

G >

"

^and/4

naG. Hence, CG

^q x, and ^so/4

nag r,.)_.

_{x, by}

F’.

^Since

qx.

^is

pseudotopological by assumption,itfollows that xa.T"

qr"-4")

_{X. Thus,}

_(X,

_q)satisfies

F.

^Theorem

4.1 nowimplies that

(X, q)

satisfies

FT.

Lemma4.4Let

J,

a,

,

^{and E}

F(J)

^beasin the statement of

RT.

^IfU

U(X)

^andU

^> (),

thenthereexists a

e U(J)

^{such that}

nag _>

naY:"^and/4

().

Proof. Let/4

U(X)

^{be suchthat/4}

> (’).

ForAff/4 and

F ,

^define

GA,F {y F (y) .

^A

^(F)},

and let

G

be any ultrafilter on

J,

with the property that

G

^is finer than the filter generated by

{GA,F"

^Ae/4andF

’}.

Since

G _> , ^{nag _> na’.}

^Also,

(GA,F)

C_

A,

for all

F e

"

^{and all}

A

/4; thus

(G)

^{=/4. |}

Theorem4.5.

For (X,

q)

PPSSI

^andT

^T, P" = ^RT.

Proof. Assume

R’.

^Let

^J,

^a,

,

^and

F(J)

beas in thestatementof

R.,

^{and assumethat} Ha.T" xforsomex EX. Let/4

U(X)

^{be such that/4}

_> (’).

By Lemma4.4, thereexistsa

e U(J)

^suchthat

na ^>_

^{Ha.7:" and}

()

^H.

Hence, na ²⁵

^{x, andso/2}

^{() qT!f4}

^{X, by}

1*.

^Sinceqr.) is pseudotopological by assumption, ^itfollows that

(’) qax--4)

_x.Thus,

_{(X, q)}

satisfies

l.

^{Theorem4.1 now}implies that

(X,

q)satisfies

RT.

Corollary^4.6. Let

(X,

q)

IPPSS[, T

CT. Then

(X, q)

^isT-regulariff,whenever p,vE

I, .T"

C

qT(,,,,)

U(X),

^and

Y

z, then

clq

^X.

Proof.

Use

the second half of the proofofTheorem 3.1 toestablish

R.’.

646 P. BROCK AND D. C. KENT REFERENCES

[1

H.J. Biesterfeldt,Jr., "Regular Convergence Spaces," Indag. Math. 28

(1966),

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[2

P.Brock andD.C.

Kent,

"ApproachSpaces, LimitTowerSpaces,and ProbabilisticConver- gencespaces," Applied CategoriedStructures

(To

appear).

[3

^C.H. Cookand H.R. Fischer, "Regular Convergence

Structure,"

^{Math. Ann. 174}

(1967),

1-7.

[4

L.C. Florescu,"ProbabilisticConvergence Structures," Aequations Math. ³⁸

(1989),

123-145.

[5

^{D.C. KentandG.D.}Richardson,"Convergence

Spaces

and Diagonal Conditions,"

Top.

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itsAppl.

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[6

^{E. LowenandR.}

^Lowen,

^{"AQuasitoposContainingCONVand}

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Intl..I. Math. andMath. Sci. II

(1988),

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^R.

^Lowen,

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A

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MET,"

Math. Nachr.

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(1989),

183-226.

[8

^{G.D.Richardsonand}D.C.

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1.Austral. Math. Soc.

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[9

B.Schweizer andA. Sklar,ProbabilisticMetricSpaces,North Holland Publ.

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NewYork, 1983.