View of Density of k-Box-Products and the existence of generalized independent families

@

^{Volume12,no.2,2011}

pp.221-225

Density of

κ

-Box-Produ ts and the existen e of

generalized independent families

Stefan Ottmar Elser

Abstra t

In this paper we will prove a slight generalisation of the Hewitt-

Mar zewski-Pondi zery theorem (theorem 2.3 below) on erning the

densityof

κ

-box-produ ts. Withthisresultwewillprovetheexisten e ofgeneralizedindependentfamiliesofbig ardinality( orollary2.5be-

low) whi hwereintrodu edbyWanjunHu.

2010 MSC:54A25,54B10;Se ondary: 03E05.

Keywords:

κ

-box-produ t,generalizedindependentfamily.

1. Introdu tion

Let

d(X)

^{denotethedensityand}

w(X)

^{the weightof the}topologi alspa e

X

^.

Denition1.1. Let

µ, κ

^{betwo ardinals with}

ℵ 0 ≤ κ ≤ µ

^and

{X i } _i∈µ

^bea

familyoftopologi alspa es.

 ^κ _i∈µ X i

^denotesthe

κ

-box-produ twhi hisindu edonthefull artesianprodu t

Q

i∈µ X _i

^{bythe anoni albase}

B =

(

\

i∈I

pr ⁻¹ _i (U i ); I ∈ P <κ (µ)

^and

U i

^isopenin

X i

)

where

P <κ (µ) := {I ⊆ µ; |I| < κ}

^.

For

κ = ℵ 0

^the

κ

-box-produ t is the usual Ty hono-produ t [8℄ and for

κ ⁺ = µ

^the

κ

-box-produ tisthefullbox-produ tmentionedbyKelley[5℄and

In this paper we will dis uss the density of

κ

-box-produ ts and the onne - tionwith innite ombinatori s. The lassi alHewitt-Mar zewski-Pondi zery

theoremstates:

d 

 ^ℵ _i∈2 ⁰ µ X _i 

≤ µ

^{forallspa es}

X _i

^with

d(X i ) ≤ µ

This has been provenfor separable spa es by E. Mar zewski [6℄ in 1941. In

1944E.S. Pondi zery[7℄provedaslightyweakerversionforHausdorspa es

andin 1947E. Hewitt[3℄provedthegeneralversionas statedabove.

Intheorem2.4wewillprove:

d ( ^κ i∈2 ^µ X _i ) ≤ µ ^<κ

^{forallspa es}

X _i

^with

d(X i ) ≤ µ

2. Density of

κ

-Box-Produ ts

Inthisse tionwewillproveageneralisationof Theorem1in[2℄. Todoso

westartwiththefollowingdenitionandproposition:

Denition2.1. Let

κ, µ

^{betwoinnite ardinalswith}

µ ≥ κ

^,

{X i } i∈I

^afamily

oftopologi alspa esandforall

i ∈ I

^let

B _i

beabaseofthetopologyon

X _i

^.

W ⊆ Q

i∈I X _i

^{is alled a}

µ

^{- ubeiffor every}

i ∈ I

^{there exists}

W _i ⊆ B ⁱ

^with

W = Q

i∈I ( T W _i )

^.

Proposition 2.2. Let

X

^{be a set,}

µ ≥ κ

^{two innite ardinals,}

{X i } i∈I

^a

familyof topologi al spa es,

{f i : X → X i } _i∈I

^{afamily offun tionsandlet}

W

beasubset of

Q

i∈I X i

^{whi h isaunionof}

µ

^{- ubes.}

Forevery ardinal

λ < κ

^{andeverytuple}

{x i } _i∈λ ; {J i } _i∈λ 

offamilies

{x i } _i∈λ ⊆ X

^and

{J i } _i∈λ ⊆ P (I)

^{, whereall}

J i

^{arepairwisedisjun tandnotempty,there}

existsasubset

Q ⊆ W

^of ardinalitylessorequalto

µ ^<κ

^{sothatforallfamilies}

{j i ; j i ∈ J i } _i∈λ

^{the following holds:}

W ∩ \

i∈λ

pr ⁻¹ _{j i} (f j i (x j i )) 6= ∅

!

⇒ Q ∩ \

i∈λ

pr ⁻¹ _{j i} (f j i (x j i )) 6= ∅

! .

Proof. Foreverytuple

{x i } _i∈λ ; {J i } _i∈λ 

with

|{i ∈ λ; |J i | > 1}| = 0

^{the laim}

isprettyobvious.

So we assume that the proposition is valid for ardinals less than

ν

^{and let}

{x i } _i∈λ ; {J i } _i∈λ 

beatuplewith

|{i ∈ λ; |J i | > 1}| = ν

^.

Withoutlossofgeneralitywemayassumethat

|J i | > 1

^forall

i ∈ ν

^and

|J i | = 1

forallother

i ≥ ν

^{andthatthereexistsatleastonefamily}

{j i ; j i ∈ J i } _i∈λ

^with

W ∩ T

i∈λ pr _j ⁻¹ _i (f j i (x j i )) 6= ∅

^.

Let

p ∈ W

^{beanpointsothat}

pr j i (p) ∈ f j i (x i )

^forall

ν ≤ i ∈ λ

^.

Thenthereexists an

J ∈ P ≤µ (I)

^with

(

q ∈ Y

i∈I

X i ; ∀j ∈ J : pr j (q) = pr j (p) )

⊆ W.

We hoose forall

i ∈ ν

^and

j i ∈ (J i − J)

^apoint

q j i ∈ f j i (x i )

^{andwedene a}

point

q ∈ W

^asfollows:

pr _i (q) :=

( pr i (p)

^,if

i ∈ (I − S

l∈ν (J l − J)) q j l

^,if

i = j l

^and

j l ∈ (J l − J)

Bythedenitionof

q

^wehave

q ∈ W ∩ T

i∈λ pr ⁻¹ _{j i} (f j i (x i ))

foreveryfamily

{j i ; j i ∈ J i } _i∈λ

^{su hthat forall}

i ∈ ν

^:

j i ∈ (J i − J)

^.

Now we haveto onsider families

{j i ; j i ∈ J i } _i∈λ

^with

j i ∈ (J i ∩ J)

^{for at}

leastone

i ∈ λ

^.

Wedene

Σ := {J _i ^∗ } _i∈ν ; |{i ∈ κ; J _i ^∗ = J i }| < ν ∧ (J _i ^∗ 6= J i ⇒ J _i ^∗ ∈ P 1 (J i ∩ J)) .

⇒ |Σ| ≤ µ ^ν ≤ µ ^λ ≤ µ ^<κ

Forall

σ = {J _i ^∗ } _i∈ν ∈ Σ

^{wedeneafamily}

{J _i ^σ } _i∈λ

^asfollows:

J _i ^σ :=

( J _i ^∗

^{, if}

i ∈ ν J i

^{, if}

i ≥ ν

For allthese

{J _i ^σ } _i∈λ

^thepropositionalreadyholds,so we an hooseaset

Q σ ⊆ W

^with

|Q σ | ≤ µ ^<κ

^{and for all families}

{j i ; j i ∈ J _i ^σ } _i∈λ

^{the following}

holds:

W ∩ \

i∈λ

pr _j ⁻¹ _i (f j i (x j i )) 6= ∅

!

⇒ Q σ ∩ \

i∈λ

pr ⁻¹ _{j i} (f j i (x j i )) 6= ∅

! .

Let

σ = {j i ; j i ∈ J i } _i∈ν

^{beafamilywith}

W ∩ T

i∈λ pr _j ⁻¹ _i (f j i (x j i )) 6= ∅

^.

Then

σ ∈ Σ

^and

Q _σ ∩ T

i∈λ pr ⁻¹ _{j i} (f j i (x j i )) 6= ∅

^.

Wedene

Q := {q} ∪ [

σ∈Σ

Q _σ

andbe ause

|Q| ≤ µ ^<κ

^{this istheset wewerelookingfor.}



Theorem2.3. Let

κ

^and

µ

^{betwoinnite ardinalswith}

µ ≥ κ

^andlet

 ^κ _i∈I X i

bea

κ

-box-produ twith

|I| ≤ 2 ^µ

^and

w(X i ) ≤ µ

^{for all}

i ∈ I

^.

Then

d(W ) ≤ µ ^<κ

^{holds for every subset}

W ⊆ Q

i∈I X i

^{whi h is a union of}

µ

^{- ubes.}

Proof. Let

|I| = 2 ^µ

^{,so wemayassumethat}

I = 2 ^µ

^.

Let

B ^∗

beabaseofthe

κ

-box-produ t

 ^κ _i∈µ D

^{ofthedis retespa e}

D = {0; 1}

with

|B ^∗ | = µ ^<κ

^.

Forall

i ∈ 2 ^µ

^let

B _i

bea base ofthe topologyon

X i

^with

|B i | = µ

^,

X

^{be a}

setwith

|X| = µ

^,

{f i ; f i : X → B i } _i∈2 µ

^{beafamilyofsurje tivefun tionsand}

ψ : 2 ^µ → Q

i∈µ D

^{beabije tion. Wedene}

Σ := {{x i } _i∈λ ; {J i } _i∈λ  ; λ < κ ∧ ∀i, j ∈ λ :

x _i ∈ X ∧ ∅ 6= J i ⊆ 2 ^µ ∧ ψ(J i ) ∈ B ^∗ ∧ (i 6= j ⇒ J i ∩ J j = ∅)}

and hoose for every

σ ∈ Σ

^{a set}

Q _σ ⊆ W

^{with all theproperties as stated}

in proposition 2.2. We dene

Q := S

σ∈Σ Q _σ

^{. Be ause of}

|B ^∗ | = µ

^{we have}

|Σ| ≤ µ ^<κ

^andtherefore

|Q| ≤ µ ^<κ

^{. Wewill nowshowthat}

Q

^isdensein

W

^.

Let

O

^{be a nonempty open set in}

W

^and

U

^{an element of the anoni-}

al base

B

of

 ^κ i∈2 ^µ X _i

^with

∅ 6= U ∩ W ⊆ O

^{. Then there exists a set}

{j i ; i ∈ λ} ∈ P <κ (2 ^µ )

^andafamily

{U i ; U i ∈ B i } _i∈λ

^with

U = T

i∈λ pr ⁻¹ _{j i} (U i )

^.

We an hooseforall

i ∈ λ

^{pairwisedisjun topensets}

B _i ^∗ ∈ B ^∗

^with

ψ(j i ) ∈ B ^∗ _i

and

x i ∈ X

^with

f j i (x i ) = U i

^.

Obviously

σ := {x i } _i∈λ ; {J i } _i∈λ 

isan element of

Σ

^{and wehavethe ondi-}

tion

∅ 6= W ∩ T

i∈λ pr _j ⁻¹ _i (f j i (x i ))

^{, thus}

Q σ ∩ U 6= ∅

⇒ Q ∩ O ⊇ O σ ∩ W ∩ U = O σ ∩ U 6= ∅

Therefore

Q

^{is densein}

W

^andwehave

d(W ) ≤ |Q| ≤ µ ^<κ

^.



ThefollowingisaslightgeneralisationoftheHewitt-Mar zewski-Pondi zery

theorem:

Theorem2.4. Let

κ

^and

λ

^{betwoinnite ardinalswith}

µ ≥ κ

^andlet

 ^κ _i∈I X _i

a

κ

-box-produ twith

|I| ≤ 2 ^µ

^and

d(X i ) ≤ µ

^{for all}

i ∈ I

^.

Then

d( ^κ _i∈I X i ) ≤ µ ^<κ .

Proof. Obviouslythere is aset

D

^{whi h is densein}

 ^κ i∈I X _i

^and

|pr i (D)| ≤ µ

forall

i ∈ I

^.

Let

 ^κ _i∈I W _i

^bethe

κ

-box-produ tofdis retespa es

W _i

^with

|W i | = µ

^and

let

f : Q

i∈I W _i → D

^{bea ontinuousandsurje tivefun tion.}

Be ause

Q

i∈I W i

^{itselfis anunionof}

µ

^{- ubesandduetotheorem 2.3thereis}

adensesubset

Q

^of

W

^with

|Q| ≤ µ ^<κ

^.

Let

O

^{beanonemptyopensetin}

 ^κ _i∈I X _i

^{. Then}

D ∩ O 6= ∅

^and

f ⁻¹ (D ∩ O)

isopenin

 ^κ _i∈I W _i

^.

So

Q ∩ f ⁻¹ (D ∩ O) 6= ∅

^and

∅ 6= f Q ∩ f ⁻¹ (D ∩ O)
⊆ f (Q) ∩ O

^.

Therefore

f (Q)

^isdensein

 ^κ _i∈I X i

^and

d  ^κ i∈I X i
≤ µ ^<κ

^.



FollowingWanjunHuwedene:

Denition 2.5. Let

S

^{beaninnite set,}

κ, λ

^and

θ

^{bethree ardinals with}

κ ≥ ℵ 0

^and

λ ≥ 2

^{. Afamily}

I = {I α } _α∈τ

^ofpartitions

I _α = I _α ^β ; β ∈ λ

of

S

is alleda

(κ, θ, λ)

-generalized independentfamily, iffollowingholds:

∀J ∈ P <κ (τ )∀f : J → λ :   

n\ I _α ^f(α) ; α ∈ J o   ≥ θ

We annowapply2.4onthistheoremandwere eivethefollowing:

Corollary 2.6. Let

κ

^and

λ

^{betwo innite ardinalswith}

µ ≥ κ

^.

On every setwith at least

µ ^<κ

^{elements existsa}

(κ, 1, µ)

-generalizedindepen- dentfamily of ardinality

2 ^µ

^.

Proof. Let

S

^{beaset of} ardinality

µ ^<κ

^.

For everyfamily

{X i } _i∈µ

^of topologi al spa eswith

d(X i ) ≤ λ

^{the following}

holdswiththeorem2.4:

d  ^κ i∈µ X _i
≤ |S|

WanjunHuprovedintheorem3.2in[4℄thatthisisequivalenttotheexisten e

ofa

(κ, 1, µ)

-generalizedindependentfamilyof ardinality

2 ^µ

^on

S

^.



A knowledgements.I amverygrateful toProf. Dr. Ulri h Felgner forhis

supportandhelpful advi e.

Referen es

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72.

[2℄ R.Engelking,Cartesianprodu tsanddyadi spa es,Fund.Math.57(1965),287304.

[3℄ E.Hewitt,Aremarkondensity hara ters,Bull.Amer.Math.So .52(1946),641643.

[4℄ W.Hu,Generalized independentfamiliesanddensesetsofBox-Produ tspa es,Appl.

Gen.Topol.7,no.2(2006),203209.

[5℄ J.L.Kelley,GeneralTopology,NewYork1955,p.107.

[6℄ E.Mar zewski,Séparabilitéetmultipli ation artésiennedesespa estopologiques,Fund.

Math.34(1937),127143.

[7℄ E. S.Pondi zery, Power problems in abstra tspa es, Duke Math. Journ. 11(1944),

835837.

[8℄ A.Ty hono,Überdietopologis heErweiterungvonRäumen,Math.Ann.102(1930),

544561.

(Re eivedDe ember2008A eptedO tober2009)

Stefan OttmarElser(stefan.elserweb.de)

Mathematis hesInstitut,EberhardKarlsUniversitätTübingen,Auf derMor-

genstelle10,72076Tübingen,Germany