WORDS AND

VOL. 21 NO. 3 (1998) 429-438

LOCAL

COMPACTNESS IN APPROACH SPACES

R.LOWEN

DepartmentofMathematics UniversityofAntwerp, RUCA

Groenenborgerlaan 171, B-2020Antwerp, Belgium C.VERBEECK

Department

ofMathematicsandComputerScience University of

Antwerp, UIA

Universiteitsplein 1, B-2610 Wilrijk, Belgium

(Received August 14, 1996)

ABSTRACT.This paper introduces some definitionsofa concept of local compactnessin ap- proach spaces. The basicrelationships between theseconceptsis studied, and measuresof local compactnessaredefined.

KEY

WORDS AND PHRASES: Approach space, measure of compactness,

(basis-)

^local compactness, measure of

(basis-)

local compactness.

1991AMS SUBJECT CLASSIFICATION CODES: 54A05;54D45.

1. INTRODUCTION

The purpose ofthispaper is to introduce andstudysomereasonabledefinitionsofaconcept of local compactnessinapproach spaces. The search fortherightnotion ifsuchauniquelydetermined generalization exists ismotivated, not only by the obviousfact that localcompactness is an important and naturalconceptintopology andhence,ashas been made clear by the development of the theoryso far, ^will be equally important in approach spaces, but more specifically it is motivatedbythe search foradescription oftheexponentialobjectsinAP.

In [1]

thisproblemwas successfullysolvedin

PRAP,

thecategoryofpre-approachspaces. Asiswell-known howeverthe situation in TOPascompared to

PRTOP,

thecategory ofpretopological spaces, isconsiderably morecomplicated, and thesamepattern presents itselfinthe theory ofapproachspaces. The link betweennotionsoflocalcompactnessand exponentialobjectin

AP

willbethetopicof forthcoming work. Unrelatedto thisproblemhowever,^wefoundthatthereare anumberofintuitivelyappealing concepts in AP which have nice properties and which even allow for quantification inthe way Kuratowski’s and Hausdorff’s measures of non-compactness quantify the topological notion of compactness.

In

this firstpartofourpaperweconcentrateon abasic study oflocal compactness concepts and their relationship.

2. PRELIMINARIES

Given a set

X

^wedenoteitspowerset by 2x and thesetofits finitesubsets by2(x). We recall those concepts andresults from

Lowen [2, 3]

which we require inthesequel.

A

map

:Xx 2x

[0, c]

iscalled adistance if it fulfils

(D1)

^VA

2X,Vx X

x

A 5(x,A)

^O.

(D2)

^Vx X:

5(x,0)

(D3) VA, B

2

x,

^Vx

X: 5(x, A

t2

B) 5(x, A)

^A

6(x, S).

(D4)

VA

2X,Vx X,

Ve

[0, o] :6(x,A) <_ 5(x,A()) +

^{e where}

A

⁽⁾ :=

{x[5(x,A) <_ e}.

If

^A

is a^collectiondistance on

^(,4(x))xex X

and^of

A

^{ideals in}C

X,

the

^[0,

function

^o]

^{x is}

6A

^called

^X

^an

-

^approach

^{[0, oo]}

^{isdefinedsystem ifbyit fulfils}

^5A(x)

^:=

^5(x,A).

(A1)

^Vx

X, V ^4(x) v(x)

^0.

(A)

^W

^{e X,} V ^{e [0,}

(V,

^N

^{]0, oc[,} 3 ^,4(x) v

^AN

^_< + e) = ^{o A(x).}

(AS)

^Vx

X, Vo A(x),

^VN

10,

3(v,),ex I] ^{A(z), Vz,}

^{y X:}

^o(y)

^AN

< x(z) + o,(y).

zEx

The elements ofan approach system are called localdistances. For ease in notation we shall, wheneverconvenientdenote anapproach system

(A(x)),ex

also simply

A.

If

A

^isanapproach systemthenA:=

(A(x))ex

^{iscalleda basisfor}

^A

^{if itfulfilstheproperties}

(bl)

^Vx

X A(x)

^{is a}basisforanideal.

(b2)

^{Vx X:}

.A(x) =/it(z)

^where

/(x)

^:=

{ [0, oc]

^x

V,N ]0,o0[,=t h(x):

^N_<

+e}.

PROPOSITION2.1

[3]

1.

If

^{4is an}approach systemon

X

thenthe map

6t" ^X

^{x 2x}

-- ^{[0, oo]" (x, A)}

^{sup inf}

^qo(a)

isadistance onX.

$.

If

^{6is a distanceonX thenthesystem}

At

^where

for

^{all x}

^X

A,(x) { [0, o01

^{x VA}

^c

^X-inf_aA

^{o(a) < 5(x,A)}}

isanapproach system onX.

3..A.

^{.,4 and}

6.a

^6.

A

set

X

equipped with anapproachsystemorequivalently^adistance iscalledan approachspace and is usuallydenoted

(X,

J

l).

^{Theassociated} distance is usually denoted simply 6 instead of

6A

^andanalogously if5isthe primary defined structure,

A

^{isusuallysimplydenoted,4,unless}

confusion mightoccur.

PROPOSITION

2.2

[3] If ^A

^{is abasis}

for

^{,4 then}

.

^{isalso obtainedby}

(,a)

^sup

If

(X, ,4)

and

(X’, 4’)

^{areapproachspaces and}

f ^{X X’}

^{is amap then}

f

^iscalledacontraction ifitfulfilsanyofthefollowing equivalent

(see

^Lowen

[3])

conditions:

(el)

^Vz

^X, Vo’ A’(f(x)) o’

^o

f

(c2)

Forany basish’for

A r,

^Vx

X, re’ ^A’(f(x))

’

^o

^{f e A(x).}

(c3)

^Vx

X, VA

C

X: ’(f(x), f(A)) <_ (x,A).

Approach spaces and contractionsformatopological construct Lowen

[3]

^{which wedenote AP.}

Forcategorical concepts, inparticular topologicalcategories, werefertoAdmek et al.

[4].

TOP canbe embeddedasabireflective and bicoreflective subconstruct of

A

P.Theembeddingsfunctor isgiven by

(X, 7")

--+

(X, AT)

^{leavingmorphisms}unaltered and where

AT

^{istheapproachsystem}

AT(x)

^:=

{o [0, c]

^x

o(x)

^0,

o

^u.s.c,at

x}

^foreveryx

^X,

^{andwhich hasas basis the}

collection

{Or

^Vneighborhood ofxfor

T}

^{for every x}

^X,

^where

^Or(x)

^{0 if x V and}

Oy(x)

^cxif x V. The associateddistanceisgiven by

6(x, A)

^{0ifx and}

6(x, A)

^if

x

.

The subconstruct thusobtained isisomorphicto

TOP.

Werecall thatthebicoreflection of

(X,A)

ⁱⁿ TOP isgivenby

idx (X,,4-a)

-+

(X,

J

I)

^where

7

^{isthetopologyon}

^X

^determined

by the neighborhood system

orequivalently bythe closureoperator :=

{x ^X (x, A) 0}

^forevery

A

2

x.

The construct

pq-MET

of extended pq-metric spaces and non-expansive maps too can be embedded as abicoreflectivesubconstruct ofAP. The embeddingis given by

(X, d) -- ^(X,)

leaving morphisms unaltered and where

Ad

^{istheapproachsystem}

J[d(x)

^:=

{o [0, ]x [o _< d(x,-)},

^Vx

e X

with obvious basis consisting of the single element

d(x, .).

^{As to be} expected the associated distance isgiven by

6(x, A) infaea ^d(x, a).

As

for topological spaces aconvergence theorycanbe developedin

AP (see E.

and

R. Lowen [5, 6]

^formore

details).

The difference withtopological spaceshowever isthatwitheach filter and eachpoint we can give a distancethepoint "is awayfrom beingalimit point"of thefilter. Precisely thisgoesasfollows. Givenaset

X, F(X)

^{istheset of}all filters on

X;

if

"

^E

^F(X),

^then

^U(’)

isthe set ofallultrafiltersfinerthan’. If C2x then

stackx

^:=

{B

^C

X

GE G

c B},

if

G

consistsofasinglesetGwewrite

stackxG

^{andifmoreoverGconsistsofa}single point a,we write

stackxa

^{forshort. If}no confusion canoccur,^wedrop the subscript X. Also,if

F

C

X

we abbreviate

U(stack F)

by

U(F).

^{Theset sec}

"

^isdefinedas the unionof all ultrafiltersfinerthan

’,

which means sec

"

^:=

^{A

^C

^X

^VFE

" ^A

^A

^{F # 0}.}

^Let

^{(X, 5)}

^{be an} approach space and

:F

F(X),

^{then thelimit}

(-function)

^{of isdefinedas}

’(z)

^{:= sup}

(f(z, A),

^{Vz X.}

AEsec

Itwillbe usefulalsotohave a descriptionof

6

^{not in termsof6 butin terms}of the associated approachsystem

t.

PROPOSITION2.3

[7]

^Let

(X, .A)

be anapproachspace. Forany

yz

_E

F(X)

^{andx EX}

wehave

AAgV(x)=

sup inf_supo(y) whereboth.4 and

.T

maybereplaced by bases.

Itis worthwile to mentionthatlimits also asin TOPprovide yeta third way to describeapproach spaces

(see

^E.andR.

^Lowen [5]

^formore

details).

^{Forusitsufficesto mention that}the distance 5can berecovered from

6

^by

(x,A)=

^inf

)lJ(x).

uu(A)

Convergence in the topological bicoreflection

(X, 7)

^ofan approach space

(X, 5)

^{can easily be}

derived from the limit

^In

^{the case} of cx)-pq-metric spaces the associatedassociatedwith5. If

" ^F(X)

^thenlimit

"

takes

-

^xon a moreⁱⁿ

^T

^{ifand only}simple and^if

^(x)

intuitive^0.

form. If

"

^E

^F(X)

^then

_Y(x)

_infsup

_{d(x, y),}

F. yF andincase

"

^{isgeneratedbyasequence}

^(x,),

^{thissimplymeansthat}

^’(x)

^limsup

^d(x,

DEFINITION 2.4 Given anapproachspace

(X,A),

^we

define

^themeasure

o.f

compactness

oI

^{X as}

#c(X)

sup inf

/(x).

uev(x)ex

The idea behind this definition is the following. Compactness means every ulrafilter should haveaconvergencepoint. Thereforetheinformationgivenby_#eisbasedonthe verification for all ultrafilterswhat are their "bes

convergence"

points. Beforeallelsewegiveanumberof equivalent forms ofthisdefinition.

PROPOSITION2.5

[2]

Foranyapproachspace

(X, riO,

we have

(X)

sup inf sup inf

(x)(z)

pEI’I=x

^{A(=)YE2(x) zEX}

sup inf supinf

(x)(z) cEl-[=x

A(x) YE2(X) zEX xEY

where

A

is abasis

for

the approach system

A.

THEOREM2.6

[2]

1.

If (X, AT)

^{is a}topological approach space then

(X, T)

^{is compact}

if

^andonly

if u(X)

^O.

2.

If (X,‘4d)

^{is an}o-pq-metric approach space then

(X,d)

^is totally bounded

if

^{and only}

if u(x) =o.

3.

If (X, ‘4d)

^{is apq-metricapproachspace then}

(X, d)

^isbounded

if

^{and only}

if #(X) ^< .

THEOREM2.7

[2]

1.

If (X,A)

^and

(X’,A’)

^are approach spaces and

f (X,A) (X’,A’)

^{is a surjective}

contractionthen

#e(X’) < #e(X).

2.

If (X, Aj)Ij

^isafamily

of

approach spaces then

3. SOME NOTIONS OF LOCAL COMPACTNESS IN

AP

In this sectionwe willdefine somenotions of localcompactness and basis-local compactness in

A P,

which ontopological spacescoincide withthe topologicalnotions of local compactness and basis-local compactness. Wewilldenote thenotions of local compactness by LCn wheren isa number between 1and5,and theassociated notionsof basis-local compactness likewise byBLCn.

DEFINITION3.1 Let

(X,,4)

beanapproachspace.

1.

(X, ,4)

^isLC1

if

^andonly

if

^itstopological

coreflection

^islocallycompact.

2.

Define

(x,.)

Lee

==

^Yx

^X,

^{VY x,3F}

^{Y #(F)}

⁰

,=,

v e x, v e v(): o(v) o,

where

])(x)

^isthe neighborhood

filter of

^{x in the}topological

coreflection.

Define

(X, .,4)

^isLC3

Yx X,.T

x-_F{"inf

(F)

⁰

== ^VxX"

_vv(x)^inf

^#c(V)=O

,=, Vx

X,

Ve

>

0,3iv

A(z),3 >

0:

({ < 6}) < .

4. (X,‘4)

^is

LC4 if

^{and only}

if

VxX,’v’e>O:

inf

(F)<e,

FeV,(x)

where

))e(x):=

stack

{{iv < }

iv

‘4(x)}.

5.

(X, ,4)

isLC5

if

^andonly

if

VxX,

Ve>0" inf

pc(F)<

inf

It iseasily verifiedthat thegiven definitionsofLC2areequivalent, andlikewise arethoseofLC3.

Wehavethefollowingobvious relations betweenthedifferentLCn.

PROPOSITION3.2 LC1 LC2 LC3

LC4

^LC5.

PROPOSITION3.3 Let

(X, T)

^{be a}topologicalspace.

Then

(X, T)

^{is locally compact}

(X, AT-)

is

LC (X, AT-)

^isLC5.

This is astraightforward result and it shows the LCn can be considered as generalizations of topological local compactnessin the context ofapproach spaces. Noticethat for atopological space,

Ve(x) V(x)

^foreverye

>

0andevery x X.

PROPOSITION3.4 1. Let

(X, d)

beanoc-pq-metric space. Then

(X, d)

^isLC2

if

^and

only

if

^{everypointpossessesa}totally bounded neighborhood.

2.

Every

c-pq-metricspaceisLC3and

LC4.

PROOF.

1. Let

(X, d)

be an x-pq-metric space. Then

(X, d)

^isLC2ifand only ifVx

X,

3V

l)(x)

#c(V)

^{0. Denotetheopen ballwith center yand radius eby}

B(y,),

then

pc(V)

^0if and onlyifVe >0, 3Y 2^(v) VC

[.Jy

B(y,

),

ifand onlyifVistotallybounded.

2. SinceLC4impliesLC3weonlyhave to showevery c-pq-metricspace

(X, d)

^isLC4. Now

(X, d)

^{is LC4} ifand onlyiffor everyx

X

and every e

>

0,

inffev,(x)pc(F) _< .

^But

B(x,e)

belongsto

])e(x)

^andby adapting the first part of theproofwesee

#c(B(x,e)) <_ .

Forevery

LCn,

wewill introduce acorrespondingnotionBLCnof basis-local compactness.

DEFINITION3.5 Let

(X,.4)

beanapproach space.

1.

(X,A)

isBLC1

if

^andonly

if

^its

TOP-coreflection

^isbasis-locally compact.

2.

(X, ,4)

isBLC2

if

^{and only}

if

Vx

X,

VV

12(x),

SW

V(x)

WCV and

#c(W)

O.

3.

(X, ‘4)

^isBLC3

if

^{and only}

if

VxX-

sup inf

#(W)=0.

VV(x)^wv() WCV

4- (X,A)

^is

BLC if

^{and only}

if

VxX, Ve>0,VFre(z),Ve’<e-

inf

#c(G) <

GE^(z) GCF

5.

(X, .,4)

^isBLG5

if

^{and only}

if

Vxfi

X,V >

^{O, VF}

V(z) V’ <

^{e" inf}

#(G) <

inf

SVe(x)(y).

GCF

Ifan approh spaceis

BLCn,

then it is alsoLCn.

In

order to see this, notice that we have

Ye(x) ,<e,(x)

for

eve

^x

^X

^and

>

0. The relationsbetween thedifferentBLCnare for theLCn:

PROPOSITION 3.6 BLC1 BLC2 BLC3 PROPOSITION3.7

Let (X, T)

^beatopologil

sce.

Then

(X, T)

bas-locally compact

(X, AT)

^BLC1

(X, Av)

^BLC2

(X, AT)

BLC3

(X,A) u BLC4 (X,A)

^isBLC5.

Thisrultillustrates theBLCncan beewed generalizations of topological bis-localcom- ptnessinthe context ofapproachspaces.

For --metric

^{spaces, we}get the following result.

PROPOSITION3.8 1. Let

(X, d)

be an -pq-metric space. Then

(X, d)

andonly i/everypointpossessesaneighborhood bcoisting

of

totally bounded neighbor- hoo&.

2.

Eve --metc

^{spaceisBLC3 and}

BLC.

For

some notionsof local compactness in

AP

we can introduceameureof local comptns whichisageneralizationinwhichall thenice properti arepreserved.

DEFINITION3.9 Let

(X, A)

^beanapproach space.

Lc3(X)

^{:= sup inf} pc(V).

x^vv()

UBZC(X)

^{:=sup sup inf}

xX V()^wv()_WCV

:=_zeX>0

( ^{v o.}

.BLc4(X)

^{:=supsup sup sup}

(

^inf

^,c(G) e’)

^VO.

GCF

U5cs(X)

^:=supsup

(

^inf

^u(F)

^inf

AV(x)(y))

^V0.

xx>0

Lcs(X)

^:=supsup_xx sup sup

(

^inf

^c(G)

^inf

^{AV,, (x)} (y)) ^v

^0.

>0FV,(x)’< ^av(,) yx

GCF

Forn

{3,

4,

5}, ()Lcn(X)

^{is called the meure}

of (b-)

local compactness ofX. This is justified by thefollongproposition.

PROPOSITION3.10

I ^(X,A)

^anapproach spaceand n

{3,

^4,

5},

then

()Lc(X)

⁰

if

^and

oy _{if (X,A) (B)L.}

The following

inualities

^{caneilybecheck.}

PROPOSITION 3.11

Let (X,A)

^{be an}appwach space. Then 1. Forall n

. [A(B)LC3(X) __ ^](B)Lc4(X) __ I(B)Lcs(X).

B

4. COUNTEREXAMPLES

Proposition3.2 gives some relationsbetween the differentnotionsof local compactnessin

A

P.

In

thissection,we willshowthese relations arereally allthere is tofind,exceptperhaps LC4

=

^LCb.

1. Notethat thereexist approach spaces which are not LC3, for instance, every topological spacewhichdoes notpossess localcompactness.

2. To prove LC2

:)

LC1,just considerthemetricspace

(d, d)

ofrational numbers with the euclidean metric. Everypointpossessesatotally bounded neighborhood, so

((, d)

^isLC2.

Butclearly the topological coreflection^isnotlocally compact,so

((, d)

isnot LC1.

3. We shallnow show LC5 LC2. Tothis end, consider the metric

(Hilbert)

^space

(12, d).

Since it is a metricspace,

(l2, d)

^{is LC4. We}already know that for allx6 and alle

>

0,

infet2 ^{Ar(x)(y) <}

^{6. We}shall prove that also

inft2 ^{Ar(x)(y) >}

^6:

infAVe(z)(y) ^{inf sup}

6(y,A)

Y6-.12 Y12 A.secVe(x)

inf sup inf

d(y,a).

Y6--12ACB(z,e)#$ a6..A

Nowtake y6

12.

Ify

9 B(x,e),

^defineA:=

{x},

^thenitisobviousthat

ACIB(x,e) :/:.

^and

infaeA d(y,a) d(y,x) >_

^6. Ify ^6_.

B(x,e) \ {x},

^{we canfind some}

^y’

⁶

B(x,e)

^{such that} d(y, y’)

_>

e. In thatcase,choose

A

^:=

{y’}.

Ify x then supanB(,e)#

infae

^A

d(x, a)

^6.

Hence

^wecanconclude

infet ^AVe(x)(y)

^e. This implies

(/2, d)

^isalso LCb. Sinceno point in

(/, d)

^hasatotally boundedneighborhood,

(/, d)

^isnotLC2.

4.

In

order toseeLC1

LC4,

considerthe following approach space

(X, A): ^{Let X}

beaset,

-

^{afilter on}

^Ag;(x)

^Xand

^{f" A(j);(x):= X [0, oo]}

^a

^f(x)

^{oo0function.ififif}

^{Y ’ ;}

^{Ifstackstackx}

^;

^{stackisaxx.filter onC}

^{g O ;}

^and

^{X ;}

^{and x}

^#

^stack6

^X,

^{x we define}

In

E.andR. Lowen

[6]

^{itwasshown that}

(X, A)

^{is an} approach space.

Moreover,

^if

H

isan ultrafilteron

X

andx6

X,

f(x) if’CHandH#stackx AH(x)

^{0o ifif}

H " ^/2

stack^andx.

^{H #}

^{stack x}

Ifthere exists an x6XsuchthatH stackx,theninf,x A/g(x) 0. If for every element x

e X, H #

^stackxand

Y

C

,

then inf,ex

AH(x) infzex f(x).

^Iffor every elementxof

X,

/4

#

^stackxand

" ^{fZ H,}

^then

^{infzx A/4(x)}

^oo.

^So

^{weget thefollowingthreecases. If}

^X

^is

finite

(i.e.,

all ultrafilters on

X

are point

filters),

^then

#c(X)

supuev(x) inf,ex

AH(X)

0.

If

X

is infinite and iffor every ultratilter

H

on

X,

not being a pointfilter,

"

^C

^H,

^then

p(X) influx f(x).

^{Finally, if}

^X

^isinfinite and there exists some ultrafilter/4 on

X,

not a point filter, such that

" ^{if H,}

^then

^#(X)

^oo.

^{Let B}

^be the intersection of all

ultrafilters on

X

which are not a pointfilter.

A

straightforward verification shows that

B {X \ ^{A A}

^is

finite}.

Usingthis

fact,

wecanstate

0 if

X

isfinite

#c(X)

inf,x

f(x)

^if

X

isinfiniteand

Y

C

{X \

^{A Ais}

finite}

c if

X

isinfiniteand9^v

{X \ ^{A A}

^is

finite}.

Inthesequelwe willsupposeXisinfinite, ^{is an}ultrafilter on

X

withonlyinfiniteelements, and

f

^o. Then for every ultrafilterL/on

X,

f(x)

^if’=L/

AL/(x)

^if

.T /2

^and

L/#

^stackx

0 ^if/,/=stackx.

Inorder to know

VE(x)

we need to find

A(x).

A(x) *

^Vbl

^U(X)

^supinf

(z) < M2(x)

U EblzEU

supin[

(z) < f(x)

^and sup inf

(z)

⁰

F^zF Ustk zU

supinf

(z) </(x)

^and

(x)

^0.

FzF

Weinfer

V(x)

^stack

{{qa

^<

e}

^{sup inf}_{F6.. z6.F}

(z) ^< f(x)

and

q(x) 0}.

Notice thatfor anyGC

X

the initialstructureonGis givenby

A(yG,I,a

^andconsequently the expressionfor

#c(G)

^isobtainedby replacingX_byGinthe expressionfor

#c(X).

^Hence

weget

inf

#c(G)

^{0 ifthere}exists somefiniteG

V(x),

inf

#e(G)

^{inf inf}

f(z)

^{if every}G

VE(x)

{C\A Aisfinite}D.’l

isinfiniteand thereexists someG

l)(x)

such that

(G \ A IA

^is

^finite}

^D

^’IG,

^and

inf

#c(G)

^{oc if every G}

Ve(x)

^isinfiniteand satisfies

aev(x)

{G \ A[A

^is

finite}

;fi

-O-

Considersomex X suchthat

f(x) < .

^Ifwetakee

^{> f(z)}

^then

^V(z)

^containsonly

infinite elements: ifqa satisfies

supF.infzFp(z < f(x)

^and

(x)

0, then for every

F "

and for everyO

>

0,there exists some z

F

such that

(z) < f(z) + g,

^so

{ < }

isinfinite, for

"

^{is an} ultrafiltercontainingonly infiniteelements. The factthat

"

^{is an}

ultrafilter also implies that

{G \ A A

is

finite} -G

^{for every G}

^V(x),

^{So we can}

conclude

infGv,(x)#,(G)

^c

>

s, whence

(X, A)

^isnotLC4.

Now choose a function

f

^{as before, but moreover satisfying}

f ^>

^0.

^For

^{every x}

^X, (z)

stackx, so

{x}

^isacompactneighborhood ofz. This means

(X, )

^isLC1.

REFERENCES

[1] LOWEN, E., LOWEN, R.,

and

VERBEECK, C.,

Exponential objectsin theconstruct

PRAP, UIA

internalreport 96-09

(1996).

[2] LOWEN, R.,

Kuratowski’s measureof non-compactness revisited,

Quart.

J. Math.

Oxford (2)

³⁹

(1988),

^235-254.

[3] LOWEN, R.,

Approach spaces: the missing link in the topology-uniformity-metric triad,

Ozford

MathematicalMonographs, Oxford University

Press (1997).

[4] AD/i.MEK, J., HERRLICH, H.,

and

STRECKER, G.E.,

Abstract and concrete categories, John Wiley

(1990).

[5] LOWEN,

E. and

LOWEN, R., A

quasitopos containing CONV and

MET

asfull subcate- gories,

Internat.

J. Math. 8J Math. Sci.3

(1988),

^417-438.

[6] ^{LOWEN, E.}

^and

^{LOWEN, R.,}

Topological quasitopos hulls of categories containingtopolog- icaland metric objects, Cah.

Top.

Gdo.

Diff.

^{Cat. 30:3}

(1989),

213-228.

[7] LOWEN, R., A

topological category suited for approximation theory?,

J.

Approximation Theory56

(1989),

^108-117.

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

This special issue will include (but not be limited to) the following topics:

• Computational methods: artificial intelligence, neu- ral networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learn- ing, multiagent learning

• Application fields: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management

• Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation

Authors should follow the Journal of Applied Mathemat- ics and Decision Sciences manuscript format described at the journal site http://www.hindawi.com/journals/jamds/.

Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Track- ing System athttp://mts.hindawi.com/, according to the fol- lowing timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Lean Yu,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;

[email protected]

Shouyang Wang,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; [email protected]

K. K. Lai,Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; [email protected]

Hindawi Publishing Corporation http://www.hindawi.com