A lter F on! is meager ifit is meager (i.e., of therst Baire ategory) in thepower-setP(!)=2 ! endowedwith the usual ompatmetrizable topology

On meager funtion spaes, network harater

and meager onvergene in topologial spaes

Taras Banakh, VolodymyrMykhaylyuk, LyubomyrZdomskyy

Abstrat.For a non-isolated point x of a topologial spae X let nw

(x) be

the smallestardinalityofa familyN ofinnitesubsets ofX suh thateah

neighborhoodO(x)X ofxontainsasetN2N.Weprovethat

eahinniteompatHausdorspaeXontainsanon-isolatedpointx

withnw(x)=0;

foreahpointx2 X withnw(x)=0 thereisaninjetivesequene

(xn)n2! inX thatF-onvergestoxforsomemeagerlterF on!;

ifafuntionallyHausdorspaeX ontainsanF-onvergent injetive

sequeneforsomemeagerlterF,thenforeverypath-onnetedspaeY

thatontainstwonon-emptyopensetswithdisjointlosures,thefuntion

spaeC

p

(X ;Y)ismeager.

Also weinvestigatepropertiesof ltersF admittingan injetiveF-onvergent

sequenein!.

Keywords:networkharater,meageronvergentsequene,meagerlter,meager

spae,funtionspae

Classiation: Primary54A20,54C35;Seondary54E52

Thispaperwasmotivatedbyaquestionoftheseondauthorwhoaskedifthe

funtionspaeC

p (!

;2)ismeager. Here!

=!n !istheremainderoftheStone-

Cehompatiationofthedisretespaeofniteordinals!and2=f0;1gisthe

doubletonendowedwiththedisretetopology. AordingtoTheorem4.1of[13℄

this questionislosely relatedto the so-alledmeageronvergene ofsequenes

in!

.

A lter F on! is meager ifit is meager (i.e., of therst Baire ategory) in

thepower-setP(!)=2

!

endowedwith the usual ompatmetrizable topology.

Thesimplestexampleofameagerlteris theFrehetlterFr=fA!:!nA

is niteg of all onite subsets of !. By the Talagrand haraterization [18℄,

a free lter F on ! is meager if and only if (F) = Fr for some nite-to-one

funtion :! !!. A funtion :!!! isnite-to-one ifforeahpointy 2!

the preimage 1

(y) is nite and non-empty. A lter F on ! is dened to be

-meager forasurjetivefuntion :!!!if(F)=Fr.

We shall say that for alter F on !, a sequene (x

n )

n2!

of points of a to-

pologial spae X F-onverges to a point x

1

2 X if for eah neighborhood

O(x

1

)X ofx

1

thesetfn2!:x

n 2O(x

1

)gbelongstothelterF. Observe

that the usual onvergene of sequenes oinides with the Fr-onvergene for

theFrehetlterFr. Thelteronvergeneof sequeneshasbeenativelystud-

iedbothinAnalysis[1℄,[4℄andTopology[5℄. Asequene(x

n )

n2!

willbealled

meager-onvergentifitisF-onvergentforsomemeagerlterFon!. Asequene

(x

n )

n2!

isalled injetive ifx

n 6=x

m

foralln6=m.

We shall prove that for a zero-dimensional Hausdor spae X the funtion

spaeC

p

(X;2)ismeagerifX ontainsaninjetivemeager-onvergentsequene.

Wereallthat atopologialspaeX is funtionally Hausdor ifforanydistint

pointsx;y 2X there isaontinuousfuntion :X !Isuhthat (x)6=(y).

Here I=[0;1℄is theunit interval. Fortopologial spaes X;Y by C

p

(X;Y)we

denotethespaeofontinuousfuntionsendowedwiththetopologyofpointwise

onvergene.

Theorem1. LetX beafuntionallyHausdorspaeandletY beatopologial

spaethat ontainstwoopen non-emptysubsetswith disjointlosures. Assume

that X is zero-dimensional or Y is path-onneted. If X ontains an injetive

meager-onvergentsequene,thenthefuntionspaeC

p

(X;Y)ismeager.

Proof: Let(x

n )

n2!

beasequenein X that F-onvergesto x

1

2X for some

meagerlterF in!. Thenthereisanite-to-onesurjetion:!!! suhthat

(F)=Fr. Byourassumption,Y ontainstwonon-emptyopensubsetsW

0

;W

1

withdisjointlosures. For everyn2!onsider thesubsetC

n

=ff 2C

p

(X;Y):

8i2f0;1g(f(x

1 )2=W

i

)8mn9k2 1

(m) (f(x

k )2=W

i ))g.

ThefatthatC

p

(X;Y)ismeagerwillfollowassoonaswehekthatC

p (X;Y)

= S

n2!

C

n

andeahset C

n

isnowheredenseinC

p (X;Y).

Toshowthat C

p

(X;Y)= S

n2!

C

n

,xanyontinuousfuntionf 2C

p (X;Y).

SineY =(Y nW

0

)[(Y nW

1

),thereisi2f0;1gsuhthatf(x

1 )2=W

i . Sine

(x

n

)isF-onvergenttox

1 andf

1

(Y nW

i

)isanopenneighborhoodofx

1 ,the

set F =fn2!:f(x

n )2=W

i

gbelongsto thelterF andthus theimage(F),

being onite in !, ontains the set fm2 ! : m ng for somen 2 !. Then

f 2C

n

bythedenition ofthesetC

n .

Next,weshow that eahset C

n

is nowhere dense in C

p

(X;Y). Fixanynon-

emptyopenset U C

p

(X;Y). Withoutlossof generality,U is abasiopenset

ofthefollowingform:

U =ff 2C

p

(X;Y):8z2Z f(z)2U

z g

for some nite set Z X and non-empty open sets U

z

Y, z 2 Z. We an

additionallyassume that x

1

2 Z. Weneed to nd a non-empty open set V

C

p

(X;Y)suhthatVUnC

n

. IfU\C

n

isempty,thenputV=U. Soweassume

that U \C

n

ontainssome funtion f

0

. Forthis funtion weannd i2f0;1g

suh that f

0 (x

1 ) 2= W

i

. Sine f

0 (x

1 ) 2 U

x

1

, we lose no generality assuming

thatU

x1

Y nW

i .

Sinethe sequene(x

n )

n2!

is injetive,weanndm n suhthat the set

X

m

=fx

k :k2

1

(m)gdoesnotintersetthenitesetZ. Chooseanyfuntion

g : Z[X

m

! Y suh that g(z)= f

0

(z) for allz 2Z and g(x) 2W

1 i forall

x2X

m .

Welaimthat thefuntion g has aontinuousextensiong:X !Y. Byour

assumption,X iszero-dimensionalor Y path-onneted. Intherstasewean

nd aretration r : X ! Z[X

m

and put g = gÆr. If Y is path-onneted,

then take any injetive funtion : g(Z[X

m

) ! Iand extend the funtion

Æg:Z[X

m

!Itoaontinuousmap:X !IusingthefuntionalHausdor

propertyof X. Sine Y ispath-onneted, themap 1

:(Æg)(Z[X

m )!Y

extends to aontinuous map : I!Y. Then theontinuous map g = Æ:

X !Y isarequiredontinuousextensionofg.

Inbothasestheset

V =ff 2C

p

(X;Y):8z2Zf(z)2U

z

; and 8x2X

m

f(x)2W

1 i g

is an open neighborhood of gthat lies in U nC

n

, witnessing that the set C

n is

nowhere densein C

p

(X;Y).

Theorem1motivatestheproblemofdetetingtopologialspaesthatontain

injetivemeager-onvergentsequenes. This will bedone for spaes ontaining

pointswithountable networkharater.

AfamilyN ofsubsetsofatopologialspaeX isalleda-network atapoint

x2X ifeahneighborhoodO(x)X ofxontainssomesetN 2N. Ifeahset

N 2N isinnite,thenN willbealledani-networkatx. Ani-networkatxexists

ifandonlyifeahneighborhoodofx in X is innite. Inthisaseletnw

(x;X)

denotethesmallestardinalityjNjofani-networkN atx. Ifsomeneighborhood

ofxinXisnite,thenletnw

(x;X)=1. IfthespaeXislearfromtheontext,

then we write nw

(x) instead of nw

(x;X) and all this ardinal the network

harater ofxinX. IfX isaT

1

-spae,thennw

(x)

0

ifandonlyifthepoint

x isnotisolated inX. Theardinalhnw

(x)=supfnw

(x;A):x2AXgis

alled thehereditary network harater at x. Pointsx 2X with hnw

(x)

0

arealledPytkeevpoints,see [11℄.

Theorem 2. If some point x of a topologial spae X has nw

(x) =

0 , then

foreahnite-to-one funtion :! !! withlim

n!1 j

1

(n)j=1 there isan

injetivesequene(x

n )

n2!

inX thatF-onvergestoxforsome-meagerlterF.

Proof: Let(N

i )

i2!

beaountable i-networkatx. SineeahsetN

i

isinnite,

weanhooseaninjetivesequene(x

k )

k 2!

in X suhthat foreveryn2! and

0i<j 1

(n)jtheset N

i

meetsthesetfx

k :k2

1

(n)g.

Itislearthatthesequene(x

n )

n2!

F-onvergesto xforthelter

F=

fn2!:x

n

2O(x)g:O(x)isaneighborhoodofxin X :

ItremainstohekthatthelterF is-meager. GivenanyneighborhoodO(x)

1

with x

k

2 O(x). Sine (N

i )

i2!

is a network at x, there is i 2 ! suh that

N

i

O(x). Takingintoaountthatlim

n!1 j

1

(n)j=1,ndn2! suhthat

j 1

(m)j>iforallmn. Nowthehoieofthesequene(x

k

)guaranteesthat

foreverymnthereis k2 1

(m)withx

k 2N

i

O(x).

Theorem2showsthatitisimportanttodetetpointsxwithountablenetwork

harater nw

(x). Let us reall that the harater (x) (resp. the -harater

(x)) ofapointx in atopologialspae X isequaltothesmallestardinality

ofaneighborhoodbase(resp.a-base)atx. A-baseatxisany-networkatx

onsistingofnon-emptyopensubsetsofX. Thesedenitionsimplythefollowing

simple:

Proposition 3. Foranynon-isolatedpointxofaT

1

-spaeX,

(1) nw

(x)(x);

(2) nw

(x)(x) provided that x hasaneighborhood ontainingno iso-

latedpointofX;

(3) nw

(x)=

0

ifxisthelimitofaninjetiveFr-onvergentsequeneinX.

Thefollowingsimpleexampleshowsthattheusualonvergeneoftheinjetive

sequene in Proposition 3(3)annot bereplaed bythe meageronvergene. It

alsoshowsthatTheorem2annotbereversed.

Example4. LetF bethemeagerlteron! onsistingof thesetsF ! suh

that

lim

n!1 jF\[2

n

;2 n+1

)j

2 n

=1:

Onthespae X =![f1gonsider thetopologyin whih allpointsn2! are

isolatedwhilethesetsF[f1g,F 2F,areneighborhoodsof1. Itislearthat

thesequenex

n

=n,n2!,F-onvergesto1inX. Ontheotherhand,asimple

diagonalargumentshowsthat nw

(1;X)>

0 .

Theorem 5. Eahinnite ompatHausdor spae X ontainsapoint x2 X

withnw

(x)=

0 .

Proof: TheoremtriviallyholdsifX ontainsanon-trivialonvergentsequene.

SoweassumethatX ontainsnonon-trivialonvergentsequene. ThenX on-

tainsalosed subset C X that admits aontinuous mapg :C !Ionto the

unit intervalI=[0;1℄, see [7, p.172℄. Replaing C by asmaller subset, we an

assumethat themap g : C !Iis irreduible, whih means that g(C 0

)6= Ifor

any proper losed subset C 0

C. Fix any ountable base B of the topology

of I. Theirreduibilityof themap g : C ! Iimpliesthat the spae C hasno

isolated points. Also the irreduibility of g implies that the ountable family

N =fg 1

(U):U 2BgofopeninnitesubsetsofCisani-networkateahpoint

x2C. Consequently,nw

(x)=

0

foreahpointx2C.

Corollary6. Foreahinnitezero-dimensionalompatHausdorspaeX and

eah topologial spae Y ontainingtwonon-empty open sets with disjoint lo-

sures the funtion spae C

p

(X;Y) is meager. In partiular, the funtion spae

C

p (!

;2)ismeager.

AlsoTheorems 2and5imply

Corollary 7. Let:!!!beanite-to-onefuntionwith lim

n!1 j

1

(n)j=

1. EahinniteompatHausdorspaeX ontainsaninjetiveF-onvergent

sequeneforsome-meagerlterF on!.

Infat,theonditionlim

n!1 j

1

(n)j=1inCorollary7annotbeweakened.

LetusreallthataninnitesubsetAisalledapseudointersetion ofafamily

ofsets F if A

F forall F 2F whereA

F meansthat AnF isnite. If

asequene (x

n )

n2!

in a topologial spae F-onvergesto apoint x

1

for some

lter F with innite pseudointersetion A !, then the subsequene (x

k )

k 2A

onvergestox

1

in thestandardsense.

Lemma 8. Let I be a ountable set and C = S

i2I C

i

, where the sets C

i are

nonempty and mutually disjoint, and sup

i2I jC

i

j < !. If H is alter on C all

of whose elements interset all but nitely many C

i

's, then H has an innite

pseudointersetion.

Proof: Thepropositionwill beprovedbyindution onn=sup

i2I jC

i

j. Inase

n=1there is nothingto prove. Suppose thatit istrue forallk <nand letI,

fC

i

: i 2 Ig, H be as abovewith maxfjC

i

j : i 2 Ig= n. If for everyH 2 H

the set fi2 I : jC

i

\Hj <ng is nite, then C itself is a pseudointersetion of

H . So suppose that J = fi2 I : jC

i

\H

0

j < ng is innite for someH

0 2 H .

In this ase we may use our indutive hypothesis for J, fC

i

\H

0

: i 2 Jg,

G =H( S

i2J C

i

\H

0

), andn 1. Thus G hasan innitepseudointersetion,

andhenesodoesH .

Proposition 9. If F is a-meager lter on! for somesurjetivefuntion :

! ! ! with lim

n!1 j

1

(n)j<1, then anysequene (x

n )

n2!

in atopologial

spaeX that F-onvergesto apoint x

1

2 X ontains asubsequene(x

n

k )

k 2!

thatonvergesto x

1 .

Proof: Choose an innite set I ! suh that sup

i2I j

1

(i)j < !. Let C

i

=

1

(i) for everyi 2 I, C = S

i2I C

i

and H =fF \C :F 2 Fg. Aording to

Lemma8thereexists aninniteset DC suhthat D

H foreveryH 2H .

Thenthesubsequene(x

i )

i2D

onvergestox

1

.

Nowletus omparetwofats:

(1) theompat Hausdorspae ! ontainsnoinjetiveFr-onvergentse-

quenes;

(2) eah inniteompatHausdor spaeX ontainsaninjetiveF-onver-

ThesetwofatssuggestaproblemofndingtheborderlinebetweenltersF that

admit an injetiveF-onvergentsequene in ! and lters that admit nosuh

sequenes. Wehopethatthisborderlinepassesnearanalytilters. Letusreall

thedenitionsofsomepropertiesoflters.

A lter F is analyti (resp. an F

-lter, F

Æ

-lter) if F is an analyti sub-

set (resp.F

-subset, F

Æ

-subset) ofthe power-set P(!)=2

!

endowed withthe

naturalompatmetrizabletopology.

A lter F is measurable (resp. null) if is it measurable (resp. has measure

zero)withrespetto theHaarmeasure ontheCantorube2

!

onsideredasthe

ountableprodutof2-elementgroups.Itiswell-knownthatalterismeasurable

ifandonlyifitisnull. Therelationsbetweenmeagerandnullltersarenottrivial

andwereinvestigatedin[18℄and[2℄. Sineeahanalytilterismeagerandnull,

wegetthefollowinghainofpropertiesoflters:

F

) analyti ) meager&null:

Wearegoing to showthat somemeagerandnulllterF admitsan injetive

F-onvergentsequene in ! while no F

-ler F admits suh a sequene. The

latterfatholdsmoregenerallyforanalytiP +

-lters.

AlterFon!isalledaP-lter(resp.aP +

-lter)ifeahountablesubfamily

CF hasapseudointersetionAthatbelongstoF (resp.to F +

). Here

F +

=fA!:8F 2FA\F 6=;g

oinideswiththeunionofallltersthatontainF. ItislearthateahP-lter

isaP +

-lter. Inpartiular,theFrehetlterF isbothaP-lterandP +

-lter.

ForalterF on! by(F)wedenoteitsharater. Itisequaltothesmallest

ardinalityjBjofthe baseB F thatgenerates F in thesense that F =fF

!:9B2B B Fg. Itiswell-knownthattheharaterofeahfreeultralteron

!isunountable.Theunountableardinalu=minf(U):U 2!n!gisalled

the ultralter number, see [3℄, [20℄. The dominating number d is the smallest

ardinalityjDjofaonal subsetD in thepartiallyorderedset (!

!

;),see[3℄,

[20℄. ByKetonen'sTheorem[10℄, eah lter F on ! with harater(F)<d is

aP +

-lter.

NowweanestablishsomepropertiesofltersFadmittinginjetiveF-onver-

gentsequenesin !.

Theorem10. AssumethatalterFadmitsaninjetiveF-onvergentsequene

(x

n )

n2!

in !.

(1) If F is aP +

-lter,then forsomesetA 2F +

thelterFjA=fF\A:

F 2FgonAisanultralter.

(2) (F)minfd;ug;

(3) F isnotananalytiP +

-lter;

(4) F isnotanF -lter.

Proof: 1. Assume that F is a P -lter. Let x

1

be the F-limit of the F-

onvergentsequene(x

n )

n2!

in!. Sinethesequene(x

n

)isinjetive,thereis

m2! suhthatforeverynm x

n 6=x

1

andheneweanxaneighborhood

U

n ofx

1

whoselosuredoesnotontainthepointx

n

. Sinethesequene(x

k )F-

onvergestox

1

, foreverynmthesetF

n

=fk2!:x

k 2U

n

gbelongstothe

lterF. Sine F isaP +

-lter, thesequene(F

n )

nm

hasapseudointersetion

A2F +

. ItfollowsfromthehoieoftheneighborhoodsU

n

thatthesetfx

n g

n2A

isdisretein!andthesequene(x

n )

n2A

isFjA-onvergenttox

1

. ByRudin's

Theorem [16℄, the map f : A ! !, f : n 7! x

n

, has injetive Stone-

Ceh

extensionf :A!!,whihimpliesthatthelterFjAisanultralter.

2. If (F)< min fd;ug, then (F) <d and by the Ketonen's Theorem [10℄

F isaP +

-lter. Bythe preeding statement, FjA is anultralter for someset

A2F +

. Consequently,

u(FjA)(F)<u

andthisisadesiredontradition.

3. IfF isananalytiP +

-lter,thenbytherststatement,FjAisanultralter

forsome subsetA 2F +

. On theother hand,the lterFjA is analyti being a

ontinuousimageoftheanalytilterF. So,FjA annotbeanultralter.

4. Assume that F isanF

-lter. Inorderto apply thepreedingstatement,

itsuÆestoshowthatF isaP +

-lter. Thisisdoneinthefollowinglemma.

Lemma11. EahF

-lterF on!is aP +

-lter.

Proof: Aording to a result of Mazur [12℄ (see also [17℄), for the F

-lter F

thereexists alowersemi-ontinuoussubmeasureonP(!)suh thatF =fA

!:(!nA)<1g. SineF6=P(!),(!)=1andthesubadditivityofimplies

that(F)=1forallF 2F. Itfollowsfrom F=fA!:(!nA)<1g that

aset A! belongsto F +

ifandonlyif(A)=1.

Toshowthat F isaP +

-lter,x any dereasingsequene ofsets (A

k )

k 2!

in

F. Let n

0

= 0 and by indution onstrut an inreasing sequene of positive

integers(n

k )

k 2!

suh that([n

k

;n

k +1 )\A

k

)>k foreveryk2!. Thentheset

A= S

k 2!

[n

k

;n

k +1 )\A

k

isapseudointersetion of(A

k )

k 2!

and belongs tothe

familyF +

as(A)=1.

Let us remark that Lemma 11annot be generalizedto F

Æ

-lters. The fol-

lowingexamplewassuggestedto theauthorsbyJonathanVerner.

Example12. Thelter

FrFr=

A!!:

n2!:fm2!:(n;m)2Ag2Fr 2Fr

on!! isanF

Æ

butnotP +

.

Lookingat Theorem10,itisnaturaltoask thefollowing

Question 13. Does ! ontain an injetive F-onvergent sequene for some

Ontheotherhand,wehavethefollowingfat:

Theorem 14. Eah innite ompat Hausdor spae X ontains an injetive

F-onvergentsequeneforsomemeagerandnulllterF.

Proof: Chooseanynite-to-onefuntion :!!! suhthat

lim

n!1 j

1

(n)j=1 and Y

n2!

(1 2 j

1

(n)j

)=0:

ByCorollary7,anyinniteompatHausdorspaeX ontainsaninjetiveF-

onvergentsequenefor some-meagerlterF. It islearthat F ismeager. It

remainsto hekthat F is null. Thelter F, being-meager, liesin theunion

S

n2!

F

n

whereF

n

=fA!:8knA\ 1

(k)6=;g. ItsuÆestoprovethat

eah set F

n

hasHaar measure zero. Observe that the set F

n

an be identied

withtheprodut Q

k n (P('

1

(k))nf;g),whihhasHaarmeasure

Y

k n 2

j' 1

(k )j

1

2 j'

1

(k )j

= Y

k n (1 2

j' 1

(k )j

)=0:

Remark 15. After writing this paper the authors learned from V. Tkahuk

that the meager property of the funtion spae C

p (!

;2) was also established

byE.G.PytkeevinhisDissertation[15,3.24℄. Gameharaterizationsoftopolo-

gialspaesX withBairefuntionspaeC

p

(X;R) weregivenin[9℄,[19℄and[14℄.

Aknowledgments. TheauthorswouldliketoexpresstheirthankstoAlanDow

andJonathanVernerforverystimulatingdisussionsandtoVladimirTkahukfor

theinformationaboutPytkeev'sresultsontheBaireategoryoffuntionspaes.

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T.Banakh:

IvanFranko NationalUniversityofLviv,Universytetska1,Lviv79000,

Ukraine

and

UniwersytetHumanistyzno-PrzyrodnizyJanaKohanowskiego, Kiele,

Poland

E-mail: tbanakhyahoo.om

URL:http://www.franko.lviv.ua/fa ult y/me hmat /Depa rtme nts/

Topology/banv.html

V.Mykhaylyuk:

DepartmentofMathematis,YuriyFedkovyhChernivtsi NationalUniver-

sity,Kotsjubynskogostr. 2,Chernivtsi58012,Ukraine

E-mail: vmykhaylyukukr.net

L.Zdomskyy:

Kurt G

odel Researh Center for Mathematial Logi, University of Vi-

enna, W

ahringer Strae25,A-1090Wien,Austria

E-mail: lzdomskylogi.univie.a.at

URL:http://www.logi.univie.a.a t/~l zdoms ky/

(Reeived Deember15,2010 , revised April23,2011 )