A NOTE ON METRIC PRESERVING FUNCTIONS

(1)

Internat.

j. Math. & Math. Sci.

VOL. 19 NO. (1996) _199-200 ¹⁹⁹

A NOTE ON METRIC PRESERVING FUNCTIONS

JOZEF DOBOS and

ZBIGNIEW PIOTROWSKI

IN’I’I{()i)U(’TION

I)I,;I,’INri’iON %%’,. ,.all I,,,,,.l,,,,i

.f

^I1{⁺ ^I1( ,,..Irlr i,r,.,.rv,,,K

,l[f(,l)

11 ,1 I! +Isanetricfor ,’t,.ryill,.Iri,",1 AI A/ Ill tl..r,.(11,/);l,i.=rl,ilr;ir) ^{IIl,’lrl,’}l,:l*’,"llllll Ill.+ l,’l ,l I.J,.,,l (,[

real. ^V,",I,.:.,t,.

I,y ,

tl,,.s,.t ,,I ;,11,,,lr=ri,r r=,,g[.,,-t,,,,,. (S ^II,,r.ik

[!].

^ll,rsik

[’2]. ’l.rl.’ [:l]

Th,.

[ollow,,,

^res,,It ^,s^,.,’11^k,,,,w,,(,.,. ^ll,,r.ik

[!].

’l’ri,,’

I’I{()I)OSI’IION I[/ ^II{⁺ ^II{ [.ii(li()i, .,liililli fly ^;,l ,)rlglil lli,’ll It llielrlr l)reserlli

TllereI&lllOllOiOlie s((lllellCeofcloe(l l)all wth ellll)()’ llilPr(’c{loll. [.

l)

We

call

1! .

,.od,ficahr,.oflh,, l’],,’l,,le;.,,,..tr,ro,,!! (.’-;,’,."l’,,ri,,,

[:lJ EXAMI’I.F.

^l)et]l.

]" IH

⁺ !!{

.f(.,-) .r. ,t ..< 2. f(.,’)

-+

^,f >^:2

i,,

J.za [4]

^it^is^show.^that

f

E

on hefoliowi=,g;property of

Foreach co.pact,^s,.l, A" lhere a ,’l,se!Iall>,"a=l II.,r. _,=i,,i,-! ,.,.l /..==,Itl:l A .C_ III ^>,’C_ I. (1

2)

(2)

200 J. DOBOS AND Z. PIOTROWSKI

MAIN

I{I,SIII.’I

.

’l’lll,()liE,,’tl I.,.!

J" ..

.,,i,I.),.,.Îlia! ^tl,,.r,, ]{) f{J’) ]I() Ji,,lil,rh Î,[ÎI,,"I,,,,I

’J’l.’. lh," lii,’tr_i)a,,

(ll{.d])

h;, fli,’ i)r,)l-’rfy (l’2)

avid

() (:)

I’II()()I" l,,’! IN I,,.

’l’hor,’for()

<

^Si,,.

W’,

.... ^"I

^/(")

W’E [.,)

/(’)

<

Tlerefore L.

EXAMPLE

Define

+ " ⁺ ^si.(

^l)

itis

.or

lilc.lto verify lat

f

^ff

.,

anl1..

rc

space

(ll{.d/)

^{Ias lh} ^prol.ry^(I2) (which yelds also^the

l[+]x-3n+ + ^]+ - ^[),

^[ ^(3n-,:l,+

^1](,,

^1.2,3, ^..) ^It ^is

^.,

dillicult toverify that

f

^and

(IR.d/)

^is metric space with 1he property

(!.1).

^{which has}

.or

theprol,crty

(1.).

lndd, the intersection of the sequence of closed ^I,all

{S,. }., (wl..re .hi.

^{has h’}

c.nre ,.

3

(2 ’’-

_{I) and the}radius

A

characterization of

e

ricpreservingf..cio.s

f

^s.ch ^{ha tho}^sl,aee^(11.^d

I)

has Lhe property

(1.1)

rern&ins anopen qulion.

A(’,KN()WI.EI)GI’]MI’NT: ’l’h,"fir,,.! .alli,.daulhorwish’s to,.xl,r,’ l,ls aplr,’cialloll ^l,otl,e l)elartlllent^of^Mat,h- elllaLics, YollllgSl,OWll

t.ae

University for their hosifilalil,y,iurin hisstay "i’h"so(’Olld llalllOd ;ttl(,hocwishes to acknowledge support, from

ungstovn

Stae Universityl{escarch(’.ou.cil

i{I’]I"I’;ltENCES

1.

BOI{SiK, J.

and

1)O!30,

Functionsvilosc co.pitionwiticv.ryetri" is ..qrc,Mat,i. Slovaca

31(1981),

3-12 (in ussian).

2.

IIORSJK,

J. and

DOBO,

()nl,etrici,roservi,gfunctions,IlealAnalysisExchtge13

(1987-88),

285-293 3. "l’il{l>]’],

F.

Metric prcscrvic(,f,cntiosn, i’roc Conf.

Tol)ology

aud

Mesurc IV

(;r’ifswald,1984,189-197.

,1.

Juza, M.

Poznfilkaofipll,’ch ll,,.lrick’ch prostor,’ch,lat.l,aticko-fyzik;il,y &gpisSAV,

(;(1956),

143-148

A NOTE ON METRIC PRESERVING FUNCTIONS

Internat.