On Spaces with Ideal Convergence Property

On Spaces with Ideal Convergence Property
Jakub Jasinski and Ireneusz Rec÷aw
Abstract. For an ideal I P (!); we continue the investigation of the class of
spaces with the I-ideal convergence property, denoted IC(I); see [2]. We show
that if I is an analytic, atomless P -ideal then IC(I) s0: If in addition I is
non-pathological and not isomorphic with Ib; then IC(I) spaces have measure
zero. We also present a characterization of the IC(I) spaces using clopen
covers.
1. Introduction
Throughout this paper X is a separable metric space and I P(!) is an ideal
on ! containing all …nite subsets of !. The power set P(!) is considered to be a
topological space with the product topology induced from 2 ! by identifying subsets
of ! with their characteristic functions. We assume that P(!) is a closed subset
of the interval [0; 1]. Recall that an ideal I P(!) is called a P -ideal if whenever
A0; A1; A3; ::: 2 I is a sequence of sets then there exists a set A1 2 I such that
An
A1 for all n < !; i.e., jAn n A1j < !: We are particularly interested in the
analytic P -ideals I P(!) because of Solecki’s theorem stating that for any such
ideal there exists a lower semicontinuous submeasure ' : P(!) ! [0; 1] such that
I = Exh(') df
= fA ! : lim '(A n n) = 0g; see [6], Theorem 3.1.
n!1
We say that a sequence of functions fn : X ! R is I-convergent to a function
g : X ! R, denoted I lim fn = g, if for every " > 0 and every x 2 X the set fn 2
! : jfn(x) g(x)j "g 2 I. In [2] authors studied spaces X where I-convergence of
sequences of continuous functions implies pointwise convergence of a subsequence
indexed by elements of a set from the dual …lter F(I) df
= fB ! : B c 2 Ig: More
speci…cally, recall
Definition 1. Let I be an ideal on ! and let X be a separable metric space.
We say that X has the I-ideal convergence property if whenever fn : X ! R is a
sequence of continuous functions I-convergent to the zero function then there exists
a set M = fm0 < m1 < m2 < :::g 2 F(I) such that for all x 2 X; limi!1 fmi (x) =
0: The class of all spaces with the I-ideal convergence property is denoted by IC(I)
and the set of all subspaces of X with the I-ideal convergence property is denoted
by ICX(I).
2000 Mathematics Subject Classi…cation. Primary 54C30, 03E35; Secondary 26A15, 40A30.
Key words and phrases. P-ideals, ideal convergence, open cover.
1

2 JAKUB JASINSKI AND IRENEUSZ REC×AW
In [2] we said that an ideal I P(!) is generated by a set C ! if I = IC
fA ! : A Cg: It is easy to show, see part ( of the proof of Proposition 2 of
[2], that for any space X; if I is generated by a single set then ICX(I) = P(X):
Ideals not generated by a a single set will be called called nontrivial.
In this note we show that it is consistent that for all analytic, nontrivial, nonpathological,
P -ideals I the class IC(I) contains countable spaces only. This result
should be viewed in context of Corollary 5 of [2] stating that under CH, for any
analytic P -ideal I there exists an uncountable space X in IC(I): Note that the class
of analytic, nontrivial, non-pathological, P -ideals is very broad and includes most
ideals discussed in literature including summable and Erd½os-Ulam ideals, see comments
following Corollary 1.9.4, p 31 of [3]. In Theorem 3 we give a characterization
of spaces with I-ideal convergence property using clopen covers.
2. Main Results
For readers’convenience we recall a few more de…nitions and simple facts. A
set X R is called a -set (or X 2 ) if every relative G subset of X is also
a relative F set in X, see [5], p 210. A mapping ' : P(!) ! [0; 1] is called a
submeasure if '(?) = 0; and '(A) '(A [ B) '(A) + '(B); see [3], p 20. A
submeasure ' is called lower semicontinuous if for all A !, lim '(A\n) = '(A):
n!1
Since R contains a closed subset homeomorphic to P(!) to simplify the notation
we assume that P(!) [0; 1] R. For any two integers n1 n2 < !; [n1; n2] df
=
fk < ! : n1 k n2g: Following the idea in the proof of Proposition 2 of [2] we
prove the following Lemma.
Lemma 1. Let I P(!)be an ideal and let X I be such that X 2 IC(I).
Then there exists a set C 2 I such that X IC: Moreover, if X = I then X = IC :
Proof. Let gn : X ! R be de…ned as follows:
gn(A) =
1 if n 2 A
0 if n =2 A
It is easy to verify that gn are continuous as both inverse images, g 1
n [f1g] and
g 1
n [f0g] are open in X: For A 2 X we have fn : gn(A) > 0g = fn : gn(A) =
1g = fn : n 2 Ag = A 2 I; so I lim gn = 0: Since:X 2 IC(I) there exists a set
M = fm0 < m1 < m2 < :::g 2 F(I) such that lim gmi(A) = 0 for all A 2 X: So for
i!1
all A 2 X; gmi (A) = 0 for su¢ ciently large i; hence for those i; mi =2 A: It follows
that M Acand consequently A M c : Take C = M c and we obtain X IC:
With C 2 I we always have IC I so if X = I then X IC I = X:
Lemma 2. If I is an analytic, nontrivial P -ideal on ! then ICR(I) does not
contain intervals.
Proof. Let ' : P(!) ! [0; 1] be a …nite lower-semicontinuous submeasure
on P(!) such that
(2.1) I = Exh(') = fA ! : lim '(A n n) = 0g;
n!1
see [6]. De…ne a descending sequence of sets fAk : k < !g as follows: A0 = !;
1
Ak = fn < ! : '(fng) k g: We shall consider the limit lim
k!1 lim
n!1 '(Ak n n):
df
=

Case 1. Assume
IDEAL CONVERGENCE PROPERTY 3
(2.2) lim
k!1 lim
n!1 '(Ak n n) = 0
but
(2.3) for every k, lim
n!1 '(Ak n n) > 0:
For every k < ! there exists a t > k such that jAk nAtj = !: Otherwise there would
be a k0 such that lim
n!1 '(Ak n n) = lim
n!1 '(Ak0 n n) for all k k0 contradicting
2.2. De…ne an increasing sequence kl, l < ! as follows: k0 = 0; kl+1 = minfk :
jAkl n Akj = !g:Set Bl = Akl n Akl+1 :
Claim. I = fA : 8ljBl\Aj < !g: If A 2 I then lim
n!1 '(Ann) = 0 so jBl\Aj < !
1
because otherwise we would have lim '(A n n)
n!1
kl+1
: On the other hand if all
sets Bl \ A are …nite then for every k there exists an n such that A n n Ak: It
follows that lim '(A n n) lim
n!1 k!1 lim
n!1 '(Ak n n) = 0 which proves the Claim:
This shows that there exists a bijection : ! ! ! 2 such that f [A] : A 2 Ig =
Ib
df
= fA ! 2 : 8n < !9m < !8k < !((n; k) 2 A ) k m)g: By Theorem 4 of
[2] it follows that ICR(I) = ICR(Ib) : Intervals are not -sets, [5], hence the
Lemma is proved in Case 1.
Case 2. Assume that
(2.4) lim
k!1 lim
n!1 '(Ak n n) = 0
and
(2.5) there exists a k0 such that lim '(Ak0 n n) = 0:
n!1
Than Ak0 2 I and I is not nontrivial because I = fA : A Ak0g: The inclusion
1
I fA : A Ak0g is clear while if jA n Ak0j = ! then lim inf'(A
n n) k0 n!1 so
A =2 I:
Case 3. Finally assume that
(2.6) lim
k!1 lim
n!1 '(Ak n n) = " > 0
Recursively de…ne a sequence fnk : k < !g as follows: n0 is such that for all n n0;
'(A0 nn) > "
2 and nk+1 k+1 be such that for all n nk+1; '(Ak+1 \[nk+1; n]) >
"
2 : Note that such nk+1exists due to the lower-semicontinuity of ': For the same
2
reason the set Z = fA : 8k > 0('(A n nk) k )g is a closed subset of P(!) and
by 2.1 Z I: Now towards a contradiction, based on Proposition 3 of [2], without
loss of generality we may assume that [0; 1] 2 ICR(I): By the same Proposition
3(2) ICR(I) also contains Z: By Lemma 1 there exists a set C 2 I such that
Z fA : A Cg: C 2 I so by 2.1 let l0 2 ! be such that '(C n m) < "
2 for
all m 2 l0 : Since for l < !, '(A 2 l +1 \ [n 2 l; n 2 l +1]) > "
2
we may select numbers
dl 2 (A 2 l +1 \ [n 2 l; n 2 l +1]) n C and de…ne the set D = fdl : l l0g: D is in…nite and
disjoint from C so clearly D * C: In particular
(2.7) D =2 Z:
Now for k > 0 let lk = maxfl0; dlog2 nkeg: We have '(D n nk)
P
'(fdlg)
l lk
P 1
: It follows that D 2 Z contradicting 2.7.
2
l lk
l +1
2
2 dlog 2 n ke
2
nk
2
k

4 JAKUB JASINSKI AND IRENEUSZ REC×AW
Remark 1. The above proof shows that if I is as in Lemma 2 and X 2 ICR(I)
then X is totally imperfect (i.e., does not contain any perfect sets.)
A subset X R is called an s0-set (or X 2 s0) if for every perfect subset
P R there exists another perfect subset Q P n X; see [5] p 217.
Theorem 1. If I is an analytic, nontrivial P -ideal on ! then ICR(I) s0:
Proof. Suppose X 2 ICR(I) and let P R be a perfect set. Let P1 P
be a perfect set homeomorphic to the Cantor set, see [7], Theorem 2.6.3. Let
h : P1 ! P1 P1be a homeomorphism, and let p : P1 P1 ! P1 be the projection,
p(x; y) = x: It is well known that there is a continuous surjection g : P1 ! [0; 1];
see [7] Theorem 2.6.13. The composition q = g p h : P1 ! [0; 1] is a continuous
surjection so by Proposition 3 of [2] q[P1 \X] 2 ICR(I): By Lemma 2 there exists a
number y 2 [0; 1] n q[P1 \ X]: It is easy to see that Q = q 1 [fyg] is a perfect subset
of P disjoint with X:
Under the Continuum Hypothesis we have an example of an nontrivial, nonanalytic
P -ideal J such that IC(J) contains large spaces.
Example 1. (CH) There exist a maximal P -ideal J such that R 2IC(J):
Proof. We will construct a sequence of subsets X 2 [!] ! ; < c: such that
X X whenever < < c. The sequence fX : < cg will be such that the
ideal J dual to the …lter F = fB
properties.
! : 9 < c(X B)g will have the desired
Let fA : < cg be a sequence of all subsets of ! and let fhfn : n < !i : < cg
be a sequence of all sequences of continuous functions fn : R ! R. Set X0 = !:
Suppose that for some < c the sets X ; are de…ned. De…ne X0 +1 as follows.
If the sequence hfn : n 2 X i is pointwise convergent to the zero function then we
set X0 +1 = X : Otherwise there exists an " > 0 and a number x 2 R such that the
set E = fn : jfn (x)j "g is an in…nite subset of X . In that case we set X0 +1 = E:
Now to obtain X +1 we consider the intersection X0 +1 \ A : If the intersection is
in…nite then let X +1 = X0 +1 \ A :Otherwise set X +1 = X0 +1 \ (! n A ):
To …nish the proof we need to de…ne the X for limit ordinals 0 < < c:
Let f n : n < !g be a sequence of ordinals co…nal in : For each n < ! let
Yn = T
X . We have Y0 Y1 Y2 ::: . and Yn are in…nite for all n < !:
k
k n
Recursively pick points xn 2 Yn n fxm : m < ng and let X = fxn : n < !g: It is
easy to verify that X X for all < .
Now having the sequence fX : < cg de…ned, it is easy to verify that the
ideal J dual to …lter F = F(J) = fB ! : 9 < c(X B)g is a maximal
P -ideal such that R 2 IC(J):
Recall that a measure on P(!) is an additive submeasure, i.e., : P(!) ! [0; 1]
is a measure if (;) = 0; (A) (A [ B); and (A [ B) = (A) + (B) for any
two disjoint sets A; B 2 P(!): Before our next theorem we would like to recall a
de…nition from [3] p 21. We say that a submeasure ' : P(!) ! [0; 1] is nonpathological
if
(2.8) '(A) = supf (A) : is a measure on P(!) and 8B !( (B) '(B)g:
An ideal I P(!) is non-pathological if I = Exh(') for some non-pathological
submeasure ': Let and be the Lebesgue measure and the outer Lebesgue

IDEAL CONVERGENCE PROPERTY 5
measure respectively. N = fY R : (Y ) = 0g and recall Ib
!9m < !8k < !((n; k) 2 A ) k m)g:
df
= fA ! 2 : 8n <
Theorem 2. If I is an analytic, non-pathological, nontrivial P -ideal not isomorphic
with Ib, then ICR(I) N .
Proof. Suppose I = Exh(') for some non-pathological submeasure ': Let
X 2 ICR(I): By Lemma 2 X is zerodimensional and we may assume that X [0; 1]:
Similarly as in the proof of Lemma 2 we de…ne a sequence of sets A0; A1; A2; ::: as
1
Ak = fn < ! : '(fng) 2k g and consider the limit 2.2. Based on the proof of
Lemma 2 our assumptions on I imply that lim
k!1 lim
n!1 '(Ak n n) = " > 0: Let fnk :
k < !g be de…ned exactly like in Case 3 of Lemma 2. De…ne Ik = Ak \[nk; nk+1 1]
and let k be a measure on P(!) such that k ' and k(Ik) > 1
2'(Ik); see 2.8.
For each k we create a partition of X into open (in X) sets fUki : i 2 Ikg such that
(Uki) < 2 k(fig)= k(Ik): We de…ne F : X ! P(!); F (x) = fi : 9k(x 2 Ukig:
Since the sets Ik are …nite and pairwise disjoint the function F is continuous and
we have F [X] 2 ICR(I) by Proposition 3(1) of [2]. Also for each x 2 X; F (x) 2 I
1
because lim '(F (x) n n) = 0: This is due to the fact that we are using
n!1 2k in the
de…nition of Ak and jF (x)\(Ak nAk+1)j 1: So with F [X] I and F [X] 2 ICR(I)
by Lemma
S
1 there
T
exists
S
a set C 2 I such that F (x) C for all x 2 X: It follows
that X
Uki: Notice that for any k < !
nni2C\Ik
[
i2C\Ik
Uki
! X
i2C\Ik
(Uki)
X 2 k(fig)=
i2C\Ik
k(Ik)
= 2 k(C \ Ik)= k(Ik)
2 k(C \ Ik)=( "
2 )
4'(C n nk)= :
Since C 2 I the last quantity converges to zero as k ! 1: It follows that
\ [
!
= 0
and (X) = 0 as well.
k>ni2C\Ik
Corollary 1. It is consistent with ZFC that if I is an analytic, non-pathological,
nontrivial P -ideal, then ICR(I) contains countable sets only.
Proof. Suppose X 2 ICR(I): If I is not isomorphic with Ib then by by Theorem
2 and Proposition 3(1) of [2] all continuous images of X are also in ICR(I)
and they have Lebesgue measure zero. Bartoszynski and Shelah, [1] showed that
consistently such spaces may be countable only. If I is isomorphic with Ib then see
Corollary 7 of [2].
Our last theorem presents a characterization of spaces with I-ideal convergence
property using clopen covers.
Uki

6 JAKUB JASINSKI AND IRENEUSZ REC×AW
Theorem 3. Suppose I is a P -ideal on ! and let X be a separable, zerodimensional
metric space. Then X 2 IC(I) if and only if for every sequence of clopen
sets U0; U1; U2; ::: X such that if for
S
any
T
x 2 X; fn : x =2 Ung 2 I; then there
exists a set A 2 F(I) such that X
Ua:
m m we have han (x) = 0; so x 2 Uan :Hence X
Ua:
m