ON THE THEOREMS OF Y. MIBU AND G. DEBS

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 3 (1996) 495-500

495

ON THE THEOREMS OF Y. MIBU AND G. DEBS

ON SEPARATE

CONTINUITY

ZBIGNIEW PIOTROWSKI

Department

ofMathematics Youngstown StateUniversity

Youngstown,

OH 44555 USA

(Received May25, 1994and in revisedform November 1,1995)

ABSTRACT. Usingagame-theoreticcharacterizationofBairespaces,conditions upon the domain and the range are given to ensure a "fat" set

C(f)

of points of continuity in the sets of type Xx

{/},

y EY for certain almost separately continuous functions

f

^{X Y Z These results} (especially Theorem B) generalize Mibu’s First Theorem, previous theorems of the author, answers one of his problemsaswellastheyarecloselyrelatedto someother results of Debs and Mibu

[2]

KEYWORI)S ANI)PItRASES:

Separate

and joint continuity,quasi-continuity, Moore spaces.

1991AMSSUBJECT CLASSIFICATION CODES: 54C08, 54C30,^26B05

I. INTRODUCTION

Sincetheappearance of the celebrated result of Namioka, many articles have beenwritten onthe topic of separate and joint continuity,seePiotrowski[3 ],forasurvey

Aside from an intensively studied Uniformization Problem- Namioka-type theorems, see Piotrowski[3],questions pertainingtoExistenceProblem

(see

below)aswellas itsgeneralizations, have beenasked

Let XandYbe"nice"(eg Polish)topological spaces, letMbemetricand let

f

^{X Y Mbe}

separatelycontinuous, thatis, continuous

wit.h

^{respecttoeach}variable whilethe otherisfixed Findthe set

C(f)

of points of(joint) continuity of

f.

Letusrecall that givenspaces

X, Y

and

Z,

and let

f

^Xx

^Y

^{Zbeafunction. For}every fixed z E

X,

the function

f ^Y

^Zdefinedby

f (t) f(x, ),

where t

Y,

iscalledan

z-secnon

of

f

A

t-section fu

^of

^f

^{is definedsimilarly.}

Onewaytoensure theexistenceof"many" points of continuityin Xx Ycanbe derivedfromthe following. Baire-Lebesgue-Kuratowski-Montgomery Theorem

(see

^Piotrowski

[3])

^Let

X

and

Y

be metricandlet

f ^{X Y R}

have all z-sections

f

continuous and have all

t-sections fu

^ofBaireclass

a Then

f

^{isof classa}

+

1

496 Z I’IOTROWSKI

Ifa 0, thatis,f,j iscontinuous,

f

^{isof class Now,}byatheorem ofBaire,

C(f)

is residual So, fweassumeaddtmnally thatXx Yis Baire, then

C(f)

is adense

Ge

subset ofX x Y ^i-I

Butonecannotrelax the assumptionspertainingtothe sectionstoomuch

EXAMPLE. LetI

[0, 1]

and letIRbe thesetofreals Put

D,

{(z,y) ^z

,

^y

,

^where

k andp are all odd numbers between0 and 2

’}

Let D t3

D,

ItiseasytoseethatC! D I

Now,

letusdefine

f ^{19- IR}

^by

^f(z,

^{y) 1, for}

^(z,

^{y) EDand}

^f(x,

^{y) 0if}

^(x,

^{y) D .allthez-}

sections

f

^{and all}the y-sections

f

^{are offirst}class of Baire and

C(f) 05

^El However,the followingthreeimportantresults hold

MIBU’S FIRST THEOREM (Mibu [2]) Let X be first countable, Y be Baire and such that Xx Yis Baire Given a metric spaceM If

f

^{Xx Y Mis}separatelycontinuous,then

C(F)

^isa dense

G

subset ofX x Y

MIBU’SSECOND THEOREM

[2].

Let

X

be secondcountable,

Y

beBaireand such that

X

x

Y

is Baire Given a metricspaceM If

f

^Xx

Y

Mhas

a) allx-sections

f

have their sets

D(f)

of points ofdiscontinuityof the first category and, b) all y-sections

f

^{arecontinuous}

Then

C(f)

is adense,

Ge

^{subsetofX xY}

Following Debs [1], a function

f"

^{X M is called}

first

^{class if for every}

^>

^{0, for every}

nonempty subset

A

C

X,

there is anonemptyset

U,

openin

A,

such thatdiana

(f(U)) ^<_ .

DEBS’ TItEOREM 1] Let XbefirstcountableYbeaspecial c-favorablespace(thusBaire),

X

xYbe Baire Givenametricspace

M

If

f" ^X

^xY

^M

^has:

a) allz-sections

f

^{offirstclass-inthe sense}of Debs and, b) ally-sections

fu

^continuous

Then

C(f)

is adense

G

^{subset of}

X

xY

2. QUASI-CONTINUITY ON PRODUCTSPACES

Afunction

f

^{X Yiscalled}quasi-continuousata point z

X

iffor each opensets

A

C

X

and

Hcf(X),

^where

zA

and

f(z)H,

^{we have}

AIntf-(H)O

^{A function}

fXY

^is

calledquast-continuous, if it isquasi-continuousateachpointzofX.

Afunction

f

^{XxY Z (X,}

^Y,

^Z arbitrary topological spaces)is said tobequasi-continuous at

(p, q) X

xYwithrespecttothevariable y, iffor every neighborhood Nof

f(p, q)

and for every neighborhoodUxVof

(p,

q),thereexists aneighborhood

V’

^ofq,with

V’

C

V,

andanonemptyopen

U’ c U,

such that forall

(z,

y)

U’

x

V’

wehave

f(z,

y) N. If

f

^isquasi-continuouswithrespect to the variable yat each pointof itsdomain, it willbecalled quasi-continuous withrespectto The definition ofa function

f

^{that is} quasi-continuous-with respect to z is quite ^similar. If

f

^{is quasi-}

continuous with respecttozandy, wesaythat

f

^issymmetrically quast-conttnuous.

One caneasily show from thedefinitions that if

f

^issynunetrically quasi-continuous,then

f

^and

fu

arequasi-continuous forall z Xand /Y. Theconversedoesnot hold.

LEMMA

(Piotrowski

[4]

Theorem4

2).

^Let

X

beaBairespace,

Y

befirstcountable and

Z

be regular If

f

^{is afunctiononXx}

^Y

^{toZ suchthat all itsz-sections}

f

are continuousand all its

-

sections

fu

^arequasi-continuous, then

f

^isquasi-continuouswithrespecttoy The conversedoesnothold

ONTHE^THEORF,MS OFY MIBI.JAND G DEBS Z97

As an immediate consequence we obtain (Piotrowski [4] ..Corollary 43) Let X and Y be first countable, Bairespacesand Zbe aregularone If

f

^{X Y Yis}separatelycontinuous, then

f

^is

symmetrically quasi-continuous

If X and Y are second countable Baire spaces ^and Z is a regular one, and a function

f:X

^{x Y}

^Z,

^then the following implications hold (which show the inclusion relations between proper classes offunctions) seeDiagram Noneof these implicationscan, ingeneralbereplaced by anequivalence,see Neubrunn[5]

f-symmetrically quasi-continuous

f-continuous

f,

fu-continuous

f,

fu-quasi-cntinuus

f-quasi-continuous

Diagram

TheBanach-Mazurtame. Wewilluse here the classicalBanach-MazurgamebetweenplayersA and

B

bothplaying withperfectinformation

(see

^Noll

[6],

Oxtoby

[7])

Astrategy forplayer

A

is a mapping ^c whosedomain is the set of all decreasing sequences

(G1, G9.,_1),

n

>

1, of nonempty opensetssuch that

c(G1, G2,-1)

is anonemptyopen^setcontained inGg.,_. Dually,astrategy for player

B

is amapping/3^whosedomain isthe setof all decreasing sequences

(U1, U2,),

n

>

0, of nonempty opensetssuch

that/3(U1, U2,)

isnonempty,openandcontained inU2, Heren 0stands for the empty sequence, for

which/3(0)

^isnonempty and open, too. Ifc, /3 ^are strategies for

A,

B respectively, ^{then the} unique sequence G1,Gg.,G3,... defined by

()=GI, a(G)=G2, /3(G1, G2)

G3,

a(G1,

G2,

G3)

G4, is called thegameof

A

withaagainst

B with/3

^Wewillsay that

A

withawinsagainst

B with/3

if ^tq

{ ^G,

^{n E}

N}

holds for thegame

G,

Gg.,... of

A

witha against

B

with/3. Conversely,we willsaythat

B with/3

^winsagainst

A

withaif

A

withadoesnotwin againstB with/3.

Wewillmakeuseof the followingtheorem,essentiallyprovedby Banach and Mazur of. Oxtoby[7], seealso Noll[6] where thegame-theoreticcharacterization ofBairespaceswasappliedto obtain some graphtheorems.

Let

E

beatopologicalspace. The followingareequivalent:

(1)

^{Eis aBaire}space;

(2)

for everystrategy/3of

B

thereexists astrategyaof

A

with winsagainst

B

with/3.

/498 Z ^PI()I’ROWSKI

3. THE MAIN RESULT

Letusrecall IfA c Xandb/isa collectionof subsetsofX,

thenst.(A,

ld)

[.J{U

U^fqA :/:

0}

Forx EX,wewrite

st.(z,N)

^nstead

ofst.({x},Lt)

Asequence

{G,,}

^{of opencoversof} X is adevelopmentofXif for eachx Xtheset

{st.(x, Gn)

n

N}

is abaseatz A developable spaceis aspacewhich has adevelopment A Moore.space^{is a}regulardevelopable space

THEOREMA. LetXbeaBairespace, Ybespace andlet

{P, },

beadevelopmentforZ If

f

^{X Y Z is}quasi-continuouswith respecttoy, then

C(f)

is adense,

G

^{subset inX} {y}, for all y E Y

PROOF. Letx X,_VEYandlet U Vbeaneighborhoodof

(x,

y) Define a strategy fora player/3in acorresponding Banach-Mazur gameplayedoverX Forthispurposeweshall order(well- ordering)thesets

X,

open neighborhoodsof yand open nonempty subsets ofX

(1)

B(O)

^{hastobedefined Since}

^Z

^{has acountable}development

,,

thereis alocal countablebaseat every point of

Z,

in particular take

{G,}

^at

f(x,)

Pick

G1

^Now by the quasi-continuity of

f

with respect to y, there is a neighborhood V of y, and a nonempty open U such that

f(U

V

c G1

^Letusfurther assume thatU and V are

thefirst

^setsintheirorderings of

X

and

Y,

respectivelywiththe above property Now,letW be thefirstnonempty opensetcontained inU andlet_xlbe thefirstelementofW Thus,W V is aneighborhood^of(x,y) So, let

(0) w

(2)

/(G,

Gg)hastobe defined, where

G,

Gg. are nonemptyopen and

G ^c G

^Now,

f

^{is quasi-}

continuous with respect to y at

(zg.,y),

pick G

3,

the first element of the base ^at f(zt,y) with

G3CG

^{Nowpick the first element U xV such that}

f(U

xV

3) C133-suchaU

^V

exists, by the quasi-continuitywith respect to yof

f

^{Now,letW3bethe first}open nonemptyset contained inU³(apriori,it canbeeventhe sameset(!))and let_z3 U³be thefirstelement of

W a.

Thus, W³x V³is aneighborhood of

(z3,

y)

So, let/(G1, G)

^W

3.

(3)

Inthiswayweproceedto define

fl

by recursion, e.,

if/3(0) G1 and/(G, G2k)

G2k+l,for all k

<

^nthentheformer stepsareavailableandwecan define

G2k+a

^{inanalogywith(2).}

(4)

Suppose

now that has beendefined. SinceXisBaire, thereis astrategyafor

A

such that

A

with awinsagainstBwith (seethedefinitionofthegame).

Let G1,

G2...

^bethegame

^A

^{withaagainstBwith}

Noticethat.

N{ o ^N} N{ o ^{N}. (3.1)}

But observe that a is winning, hence this intersection is nonempty; i.e, x* 5

I’I{W,,

^:n

N},

^so (x’,y)

(U V)

t2

(X {y}).

This in turn shows the density of

C(f)

in

X {y}

The

G6

^part

follows easily from the construction I-I

A

spacewillbecalledquast-regularif foreverynonemptyopenset

U,

there isanonemptyopenset Vsuch thatCI V

c

U Obviously, every regularspaceisquasi-regular.

Let

.A

be an open covering ofa spaceX Then a subset S ofX is said to be A-smallifS is containedin a member of^.,4 AspaceXis saidtobestronglycountably completeifthere is asequence

()N l’lll:.TI1EORI.MS^()FY M1BIJ^ANI)G I)EBS 499

{Ai 1, 2 ofopen coverings ofX such that a sequence

{F,}

ofclosed subsets ofX has a nonempty intersectionprovidedthatF, F,. for all and eachF, is.A,-small

The class of strongly countably complete spaces includes locally countable compact spaces and completemetricspaces

In view of the following (Piotrowski [8], Theorem 46 see also Lemma ³ of Piotrowsk [9]) Everyquasi-regular,

strongJy

countably complete spaceXis a Bairespace

TheoremAis a stronggeneralizationof thefollowing

(Piotrowski[8],Theorem 45) LetXbeaspace, Ybequasi-regular, strongly countably complete andZbemetric If

f

^{X Y Zis}quasi-continuouswithrespecttoz, then forallzEXthesetof points of joint continuity of

f

^{is adense}

G

^of

{z}

^Y Further, observe thatourTheoremAanswers, inpositive,thefollowing

(Piotrowski [8], ^Problem4 11) Does Theorem4 5(of

[8])

^holdifYisonlyassumedtobeaquasi- regular^Bairespace9

ThefollowingTheoremBisthemainresult ofthispaperand itsproof easily^followsfromthelemma and TheoremA

THEOREM B. Let

X

befirstcountable,

Y

beBaireand

Z

beMoore If

f ^X

^{x Y}

^Z

^{has all}

its z-sections

fx

quasi-continuous and all its

v-sections fy

continuous, then for allz EX, the set of points of continuity of

f

^{is adense}

G

^{subsetof z Y}

The above result strongly generalizes(see the assumptions upon Y and Z) the followingknown theorem

(Piotrowski [8],^{Theorem4 8 seealsoTheorem5 ofPiotrowski}[9]) Let Xbe firstcountable,Ybe strongly countably complete, quasi-regular ^and Z beametric space If

f

^{X Y}

^Z

is a function such that all its z-sections

fx

^arequasi-continuous and all its

v-sections fy

are continuous,thenfor all z

X,

thesetof points ofjoint continuity of

f

^{is adenseG,subset of z}

^}

^Y

OurTheoremBgeneralizesinmany waysMibu’sFirstTheorem-seeIntroduction

Itisalsoclosely relatedtoMibu’s SecondTheoremand Debs’ Theorem ibidem Observethough, that quasi-continuity of a function does not imply nor is implied, by the condition of being of first class- in the senseofDebs

Really,let

f ^{[0, 1]}

^]Rbe given by

f(z)

0, ifz

#- 1/2.

^{Thensuchafunction}

f

^{isoffirstclass,in}

thesenseof Debsand, clearly,it isnotquasi-continuous

There are quasi-continuous ^functions

f:li-

^R which are of arbitrary class ofBaire or not Lebesgue measurable-see Neubrunn[5]for more details

REMARK 1. The studies of thecontinuity points offunctionswhose ranges arenotnecessarily metric have been done already in the 1960’s, see Klee and Schwarz [10] or later in the 1980’s, see Dubins 11],weomit hereanextensive literatureofthisapproach,when the rangeis auniformspace

REMARK2. Recently, the author has obtained some results ofthispaper using though entirely differenttechniques,seePiotrowski 12]

ACKNOWLEDGMENT.

^{The authorwishes toexpresshisgratitudeto arefereewhosecommentsand}

remarks have improved thepresentation^{oftheresults Also,}thanks are duetotheResearchCouncil of

Youngstown

StateUniversity foragrantwhichenabled the authortocompletethisresearch

500 Z ^PI()lROWSKI

REFERENCES

[1] DEBS, G,Fonctions

separment

continuesetdepremiere classesurespace produit, Math. Stand 59( 86), ^{22- 30}

[2] MIBU, Y, On quasi-continuous mappings defined on product spaces,

Proc..Japan

Acad. 192 (1958), ^189-192

[3] PIOTROWSKI,

Z,

Separateand joint continuity,RealAnalystsExchange,11(1985),293-322

[4]

PIOTROWSKI, Z, Quasi-continuity and product spaces, Proc. Intern.

(’ate.

Geom. "lop., Warsawa (1980),349-352

[5] NEUBRUNN, T,Quasi-continuity, Real AnalysisbScchange, 14

(1988-89),

^259-306 [6] NOLL, D,Bairespacesand graph theorems,Proc. Amer.Math.

Sac.,

96

(1986),

^141-151

[7] OXTOBY, J C, The Banach-Mazur game and Banach category theorem, Contribution to the Theory

of

^{Games, Vol 3, Ann.}

of

Math. Studies, No 39, Princeton Univ Press, Princeton, NJ, 1957

[8] PIOTROWSKI, Z,

A studyofcertainclassesof almost continuous functionsontopological spaces, Ph D. Thesis,Wroclaw, 1979

[9] PIOTROWSKI,

Z,

Separatealmostcontinuity andjoint continuity, Colloquia MathematlcaJdmos Bolyat23 Topology,Budapest(Hungary)(1978),^957-962

[1)] KLEE, V,

Stability of the fixed point property,Colloq.Math. $(1961),^43-46.

[11]

DUBINS, L E and

SCHWARZ,

G.,Equidiscontinuity of Borsuk-Ulam functions,

Pacific

^J.Math.,

95(1981), 51-59

[12] PIOTROWSKI, Z,

Mibu-type theorems, Classtcal Analysts, Proc. Intern.

Conf. ^WSI,

^Radom

(1994), ^133-139