Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

0.26 MB·

by  Boaz Tsaban

HEREDITARILY HUREWICZ SPACES AND
ARHANGEL’SKI˘I SHEAF AMALGAMATIONS
2
BOAZ TSABAN ANDLYUBOMYRZDOMSKYY
1
0
2 Abstract. A classical theorem of Hurewicz characterizes spaces with

n the Hurewicz covering property as those having bounded continuous
a images intheBaire space. Wegiveasimilar characterization forspaces
J X which havetheHurewicz property hereditarily.

3 WeproceedtoconsidertheclassofArhangel’ski˘ıα1spaces,forwhich
everysheafatapointcanbeamalgamatedinanaturalway. LetCp(X)
] denote the space of continuous real-valued functions on X with the
N
topology of pointwise convergence. Our main result is that Cp(X) is
G an α1 space if, and only if, each Borel image of X in the Baire space
. is bounded. Using this characterization, we solve a variety of problems
h
posed in theliterature concerning spaces of continuousfunctions.
t
a
m
[

2
v
1
1 Contents
2
0 1. Introduction 2
. 2. A characterization of hereditarily Hurewicz spaces 3
4
0 3. Bounded Borel images 6
0 4. Applications 11
1
4.1. QN spaces 11
:
v 4.2. Convergent sequences of Borel functions 12
i
X 4.3. Almost continuous functions 13
r 4.4. wQN spaces and the Scheepers Conjecture 13
a
4.5. QN spaces and M spaces 14
4.6. wQN spaces 15
∗
4.7. Bounded-ideal convergence spaces 15
4.8. Bounded Baire-class α images 17
5. Closing the circle: Continuous bounded images again 17
Acknowledgments 19
References 19
1991 Mathematics Subject Classification. Primary: 37F20; Secondary 26A03, 03E75.
Keywordsandphrases. pointwiseconvergence,point-cofinitecovers,α1,eventualdom-
inance, Hurewicz property,selection principles, QN sets, ideal convergence.
Partially supported by theKoshland Center for Basic Research.
1
2 BOAZTSABANANDLYUBOMYRZDOMSKYY
1. Introduction
We are mainly concerned with spaces X which are (homeomorphic to)
sets of irrational numbers, and we recommend adopting this restriction for
clarity. Our results (and proofs) apply to all topological spaces X in which
each open set is a union of countably many clopen sets, and the spaces
considered are assumed to have this property.1
Fix a topological space X. Let A,B be families of covers of X. The
space X may or may not have the following property [35].
U (A,B): Whenever U ,U ,··· ∈ A and none contains a finite sub-
fin 1 2
cover, there exist finite sets F ⊆ U , n ∈ N, such that { F : n ∈
n n n
N} ∈ B.
S
Let O denote the collection of all countable open covers of X.2 A cover U
of X is point-cofinite if U is infinite and each x ∈ X, is a member of all
but finitely many members of U.3 Let Γ denote the collection of all open
point-cofinite covers of X. Motivated by studies of Menger [26], Hurewicz
[19] introduced the Hurewicz property U (O,Γ).
fin
Hurewicz [19] essentially obtained the following combinatorial charac-
terization of U (O,Γ) (see Rec law [29]). For f,g ∈ NN, f ≤∗ g means
fin
f(n) ≤ g(n) for all but finitely many n. A subset Y of NN is bounded if
there is g ∈ NN such that f ≤∗ g for all f ∈ Y.
Theorem 1 (Hurewicz). X satisfies U (O,Γ) if, and only if, every con-
fin
tinuous image of X in NN is bounded.
This characterization has found numerous applications—see [38, 24, 41]
and references therein. We give a similar characterization for hereditarily
Hurewicz spaces, that is, spaces X such that each subspace of X satisfies
U (O,Γ).
fin
The property of being hereditarily Hurewicz was studied in, e.g., [15, 28,
27]. Rubinintroducedapropertyof subsetsof Rsuchthattheexistence ofa
setwiththispropertyisequivalenttothepossibilityofacertainconstruction
ofbooleanalgebras[31]. Miller[27]provedthattheRubin spaces areexactly
the hereditarily Hurewicz spaces.
The property of being hereditarily Hurewicz also manifests itself as fol-
lows: A set X ⊆ R is a σ′ space [32] if for each F set E, there is an F set
σ σ
F such that E∩F = ∅ and X ⊆ E∪F. This property was effectively used
in studies of generalized metric spaces [13]. Recently, Sakai proved that X
is a σ′ space if, and only if, X is hereditarily Hurewicz (Theorem 6 below).
There exist additional classes of hereditarily Hurewicz spaces in the liter-
ature. We describe some of them.
1Every perfectly normal space (open sets are Fσ) with upper inductive dimension 0
(disjoint closed sets can be separated by a clopen set) has the required property. Thus,
thespaces considered in thereferences also havethe required property.
2IfX is Lindel¨of, we can consider arbitrary open covers of X.
3Traditionally, point-cofinite covers were called γ-covers [17].
HUREWICZ AND SHEAF AMALGAMATIONS 3
A topological space is Fr´echet if each point in the closure of a subset of
thespace is a limit of aconvergent sequence of points from that subset. The
following concepts, dueto Arhangel’ski˘ı [1, 2], are important in determining
when a product of Fr´echet spaces is Fr´echet. Let Y be a general topological
space (not necessarily Lindel¨of or zero-dimensional). A sheaf at a point
y ∈ Y is a family of sequences, each converging to y. To avoid trivialities,
we consider only sequences of distinct elements. We say that a countable
set A converges to y if some (equivalently, each) bijective enumeration of A
converges to y. Thespace Y is an α space if for each y ∈Y, each countable
1
sheaf {A : n ∈ N} at y can be amalgamated as follows: There are cofinite
n
subsets B ⊆ A , n ∈ N, such that the set B = B converges to y. The
n n n n
references dealing withα spaces are too numeroustobelisted here; see [40]
1
S
and the references therein for a partial list.
Fix a space X. The space C (X) is the family of all continuous real-
p
valued functions on X, viewed as a subspace of the Tychonoff product RX.
A sequence of results by Bukovsky´–Rec law–Repicky´ [10], Rec law [30], Sakai
[33], and Bukovsky´–Haleˇs [9], culminated in the result that if C (X) is an
p
α space, then X is hereditarily Hurewicz. Our main result is that if C (X)
1 p
is an α space, then each Borel image of X in NN is bounded. It is easy
1
to see that the converse implication also holds, and we obtain a powerful
characterization of spaces X such that C (X) is an α space.
p 1
Historically, the realization that if C (X) is an α space then X is hered-
p 1
itarily Hurewicz goes through QN spaces [10]: Let Y be a metric space. A
function f : X → Y is a quasi-normal limit of functions f : X → Y if
n
there are positive reals ǫ , n ∈ N, converging to 0 such that for each x ∈ X,
n
d(f (x),f(x)) < ǫ for all but finitely many n. A topological space X is
n n
a QN space if whenever 0 is a pointwise limit of a sequence of continuous
real-valued functions on X, we have that 0 is a quasi-normal limit of the
same sequence. QN spaces are studied in, e.g., [10, 30, 36, 28, 11, 33, 9].
In [33, 9] it was shown that X is a QN space if, and only if, C (X) is an
p
α space. Thus, QN spaces are also characterized by having bounded Borel
1
images in NN.
We use our main theorem to show that quite a few additional properties
studied in the literature are equivalent to having bounded Borel images in
NN, and consequently solve a variety of problems posed in the literature. To
make the paper self-contained and accessible to a wide audience, we supply
proofs for all needed results. Often, our proofs of known results are slightly
simpler than those available in the literature.
2. A characterization of hereditarily Hurewicz spaces
N
Let N = N∪{∞} be the one-point compactification of N, and endow N
N
withtheTychonoffproducttopology. Anelementf ∈ N iseventually finite
if there is m such that f(n) < ∞ for all n ≥ m. Let EF be the subspace of
4 BOAZTSABANANDLYUBOMYRZDOMSKYY
N N
N consisting of all eventually finite elements of N . The partial order ≤∗
extends to EF in the natural way.
Theorem 2. X is hereditarily U (O,Γ) if, and only if, every continuous
fin
image of X in EF is bounded.
Proof. (⇒) Assume that Ψ : X → EF is continuous. For each n, define the
following (G ) subset of Ψ[X]:
δ
G = {f ∈ Ψ[X] : (∀m ≥n) f(m)< ∞}.
n
Let Tn : G → NN be the shift transformation defined by Tn(f)(m) =
n
f(m+n) for all m.
For each n, X = Ψ−1[G ] ⊆ X, and therefore satisfies U (O,Γ). By
n n fin
Theorem 1, Tn[Ψ[X ]] = Tn[G ] is a bounded subset of NN. Thus, G is a
n n n
bounded subset of EF, and therefore so is Ψ[X] = G .
n n
(⇐) First, note that Hurewicz’s Theorem 1 and our assumption on X
imply that X satisfies U (O,Γ). S
fin
Lemma 3. If each G subset of X satisfies U (O,Γ), then X is hereditarily
δ fin
U (O,Γ).
fin
Proof. Let Y ⊆ X. Assume that U , n ∈ N, are covers of Y by open subsets
n
of X, which do not contain finite subcovers. Each U is an open cover of
n
G = U ⊇ Y, and has no finite subcover of G. As G is a G subset of
n n δ
X, it satisfies U (O,Γ). Thus, there are finite F ⊆ U , n ∈ N, such that
fin n n
{ FT:nS∈ N} is a point-cofinite cover of G, and therefore of Y. (cid:3)
n
SAssume that G is a Gδ subset of X.
Lemma 4 (Sakai [33]). For each G subset G of X, there is an open point-
δ
cofinite cover {U :n ∈ N} of X such that G = U .
n n n
Proof. Gc = nCn with each Cn closed. If A isTclosed and B is open, then
B is a union of countably many disjoint clopen sets, and therefore A∩B is a
S
union of countably many disjoint closed sets. Thus, each of the disjoint sets
C \(C ∪...,C ), n ∈ N, is a union of countably many disjoint closed
n 1 n−1
sets. Hence, Gc = C˜ where the sets C˜ are closed and disjoint, and
n n n
therefore G = C˜c, where {C˜c : n ∈ N} is an open point-cofinite cover of
n nS n
X. (cid:3)
T
So, let {U : n ∈ N} be an open point-cofinite cover of X such that
n
G = U . For each n, let U = Cn, a union of disjoint clopen sets.
n n n m m
Define Ψ : X → EF by
T S
m m ∈ N,x ∈ Cn
Ψ(x)(n) = m
(∞ x ∈/ Un
As {U : n ∈ N} is a point-cofinite cover of X, Ψ(x) is eventually finite for
n
each x ∈ X.
HUREWICZ AND SHEAF AMALGAMATIONS 5
The function Ψ is continuous: A basic open set in EF has the form V
n n
such that there are finite I ,I ⊆ N and elements m , n ∈ I ∪I , for which:
0 1 n 0 1
Q
For each n ∈ I , V = {m }, for each n ∈ I , V = {m ,m +1,...}∪{∞},
0 n n 1 n n n
and for each n ∈/ I ∪I , V = N∪{∞}. Now,
0 1 n
Ψ−1 V = Cn ∩ X \ Cn
n mn   k
"nY∈N # n\∈I0 n\∈I1 k<[mn
is open.   
Thus, Ψ[X] is bounded by some g ∈ NN. Now,
G = {x ∈ X : (∀n) Ψ(x)(n) < ∞}= Ψ−1[{f ∈ NN :f ≤∗ g}].
The set {f ∈ NN : f ≤∗ g} is an F subset of EF. Indeed, let {g : n ∈
σ n
N} enumerate all elements of NN which are eventually equal to g. Then
{f ∈ NN : f ≤∗ g} = {f ∈ EF : f ≤ g }. Thus, G is an F subset of
n n σ
X. As U (O,Γ) is hereditary for closed subsets and preserved by countable
fin
unions, G satisfies U (SO,Γ). (cid:3)
fin
Recall that a topological space X is a σ space if each G subset of X is
δ
an F subset of X. The proof of Theorem 2 actually shows that (2 ⇒ 3),
σ
(2 ⇒ 1), (3 ⇒ 4), and (4 ⇒ 2) in the following theorem (and therefore
establishes it).
Theorem 5. The following are equivalent:
(1) X is hereditarily U (O,Γ).
fin
(2) Each G subset of X satisfies U (O,Γ).
δ fin
(3) Every continuous image of X in EF is bounded.
(4) X satisfies U (O,Γ) and is a σ space. (cid:3)
fin
The implication (1 ⇒ 4) in Theorem 5 was first proved by Fremlin and
Miller [15]. The implication (4 ⇒ 1) can be alternatively deduced from
Theorem 3.12 of [11] and Corollary 10 of [8]. An additional equivalent
formulation was discovered by Sakai. Recall the definition of σ′ space from
the introduction (page 2).
Theorem 6 (Sakai). Let X ⊆ R. X is a σ′ space if, and only if, X is
hereditarily U (O,Γ).
fin
Proof. This follows from Theorem 5.7 of [22]: X satisfies U (O,Γ) if, and
fin
only if, for each G set G ⊆ R containing X, there is an F set F ⊆ R such
δ σ
that X ⊆ F ⊆ G.
(⇒) As being a σ′ space is hereditary, it suffices to show that X satisfies
U (O,Γ). Indeed, for each G set G ⊆ R containing X, let E = R \ G,
fin δ
and take an F set F ⊆ R disjoint from E such that X ⊆ E ∪ F. Then
σ
X ⊆ F ⊆ G.
(⇐) Let E ⊆ R be F . As X\E satisfies U (O,Γ) and is a subset of the
σ fin
G set R\E, there is an F set F ⊆ R such that X\E ⊆ F ⊆R\E. Then
δ σ
E∩F = ∅ and X ⊆ E ∪F. (cid:3)
6 BOAZTSABANANDLYUBOMYRZDOMSKYY
ToindicatethepotentialusefulnessofTheorem2,weuseittogiveslightly
moredirectproofsoftwoknowntheorems. RecallthedefinitionofQNspaces
from the introduction (page 3).
Theorem 7 (Rec law [30]). If X is a QN space, then X is hereditarily
U (O,Γ).
fin
Proof. Let Y ⊆ EF be a continuous image of X. Then Y is a QN space.
By Theorem 2, it suffices to show that Y is bounded. For each n, and each
y ∈ Y, define
1
f (y) = ,
n
miny−1(n)
using the natural conventions that min∅= ∞ and 1/∞ = 0. lim f (y) = 0
n n
for all y ∈ Y. As Y is a QN space, there are positive ǫ , n ∈N, dominating
n
this convergence. For each k, let
Y = {y ∈ Y :(∀n ≥ k) f (y) < ǫ }.
k n n
Y = Y . We will show that each Y is bounded.
k k k
Fix k. Take an increasing g ∈ NN such that g(1) = k and for each n,
S
ǫ < 1/n for all m ≥ g(n). Then Y is bounded by g: Let y ∈ Y . Fix
m k k
n such that y(n) < ∞. If y(n) ≤ k, then y(n) ≤ g(1) ≤ g(n). Otherwise,
y(n)> k, and since y ∈ Y , f (y) <ǫ . Thus,
k y(n) y(n)
1 1
≤ = f (y) < ǫ ,
n miny−1(y(n)) y(n) y(n)
and therefore y(n) cannot be greater than g(n). (cid:3)
Theorem 8 (Rec law [30]). If X is QN space, then X is a σ space.
Proof. Theorems 5 and 7. (cid:3)
The following sections give a deeper reason for the last two theorems.
3. Bounded Borel images
Our main goal in this section is to establish the equivalence in the follow-
ing Theorem 9. The implication (2 ⇒ 1) in this theorem is Proposition 9 of
Scheepers [36]. The implication (1 ⇒ 2) is the more difficult one, and will
be proved in a sequence of related results.
Theorem 9. The following are equivalent:
(1) C (X) is an α space.
p 1
(2) Each Borel image of X in NN is bounded.
Proof. (2 ⇒ 1) Consider a sheaf {A : n ∈ N} at f ∈ C (X). For each
n p
n, enumerate A = {fn : m ∈ N} bijectively. Define a Borel function
n m
Ψ :X → NN by
Ψ(x)(n) = min{k :(∀m ≥ k) |fn(x)−f(x)|≤ 1/n}.
m
HUREWICZ AND SHEAF AMALGAMATIONS 7
Let g ∈ NN bound Ψ[X], and take the amalgamation B = {fn : m ≥
n m
g(n)}. Then B converges to f.
S
(1 ⇒ 2)AssumethatC (X)isanα space. ThenthesubspaceC (X,{0,1})
p 1 p
of C (X), consisting of all continuous functions f : X → {0,1}, is an α
p 1
space.4 Each element of C (X,{0,1}) has the form χ , the characteristic
p U
functionof aclopensetU ⊆ X. Immediately fromthedefinition,asequence
χ of elements of C (X,{0,1}) converges pointwise to the constant func-
Un p
tion 1 if, and only if, {U : n ∈ N} is a clopen point-cofinite cover of X.
n
This gives the following, which is due to Bukovsky´–Haleˇs (cf. [9, Theorem
17]), and independently Sakai (cf. [33, Theorem 3.7]).
Lemma 10. The following are equivalent:
(1) C (X,{0,1}) is an α space;
p 1
(2) For each family {U : n ∈ N} of pairwise disjoint clopen point-
n
cofinite covers of X, there are cofinite V ⊆ U , n ∈ N, such that
n n
V is a point-cofinite cover of X. (cid:3)
n n
A funSction f with domain X is a discrete limit of functions fn, n ∈ N, if
for each x ∈X, f (x) = f(x) for all but finitely many n.
n
Each bijectively enumerated family U = {U : n ∈ N} of subsets of a set
n
X induces a Marczewski map U :X → P(N) defined by
U(x) = {n ∈ N :x ∈U }
n
for each x ∈X. The main step in our proof is the following.
Lemma 11. Assume that C (X,{0,1}) is an α space, and U = {U :
p 1 n
n ∈ N} is a bijectively enumerated family of open subsets of X. Then the
Marczewski map U :X → P(N) is a discrete limit of continuous functions.
Proof. First, consider the case where for each n, U is not clopen.
n
For each n, write U as a union Cn of nonempty disjoint clopen
n m m
sets. We may assume that the partitions are disjoint: Inductively, for each
n = 2,3,..., consider the elements CSn, m ∈ N, of the nth partition. For
m
each m, if Cn appears in the partition of U for some k < n, merge (in the
m k
nth partition) Cn with some other element of the nth partition. Continue
m
inthismanneruntilthenthpartition is disjointfromall previouspartitions.
Thus, the families U = {(Cn)c : m ∈ N} are disjoint clopen point-
n m
cofinite covers of X. By Lemma 10, there are k , n ∈ N, and subsets
n
V = {(Cn)c : m ≥ k } ⊆ U , n ∈ N, such that V is a point-cofinite
n m n n n n
cover of X. In other words,
S
∞
V = (Cn)c : n∈ N
 m 
m\=kn 
is a point-cofinite cover of X.
 
4Infact,bythemethodsofGerlits–Nagy[18],theconverseimplicationalsoholds. This
fact will not beused in our proof.
8 BOAZTSABANANDLYUBOMYRZDOMSKYY
For each n,m, let
max{m,kn}
Un = Cn.
m i
i=1
[
For each m, define Ψ :X → P(N) by
m
Ψ (x) ={n :x ∈ Un}.
m m
As each Un is clopen, Ψ is continuous. It remains to prove that, viewed
m m
as a Marczewski map, U is a discrete limit of the maps Ψ , m ∈ N.
m
Fix x ∈ X. Let N be such that x ∈ ∞ (Cn)c for all n ≥ N. For each
m=kn m
n < N with x ∈ U , let m be such that x ∈ Un . Set M = max{m :n <
n n T mn n
N}.
Fix m ≥ M. We show that x ∈ Un if, and only if, x ∈ U . One
m n
direction follows from Un ⊆ U . To prove the other direction, assume that
m n
x ∈ U , and consider the two possible cases: If n < N, then x ∈ Un
n m
because m ≥ M ≥ m , and we are done. Thus, assume that n ≥ N. Then
n
x ∈ ∞ (Cn)c. As x ∈U = Cn, it follows that x ∈ kn−1Cn ⊆ Un.
i=kn i n m m i=1 i m
Thus, for each x ∈X there is M such that Ψ (x)= U(x) for all m ≥ M.
m
T S S
This completes the proof in the case that no U is clopen.
n
For the remaining case, let I ⊆ N be the set of all n such that U is
n
not clopen. The previous case shows that U = {U : n ∈ I}, viewed as
I n
a Marczewski function from X to P(I), is a discrete limit of continuous
functions Ψ : X → P(I).
m
For each m, define Φ : X → P(N) by
m
Φ (x) = {n :(n ∈I and n∈ Ψ (x)) or (n ∈/ I and x ∈U )}.
m m n
Then U is a discrete limit of the continuous functions Φ , m ∈N. (cid:3)
m
As X satisfies item (2) of Lemma 10, it satisfies U (O,Γ): Refine each
fin
given cover to a clopen cover, turn it to a clopen point-cofinite cover by
taking finite unions, and make the point-cofinite covers disjoint.
Assume that U is a countable family of open subset of X. By Lemma
11, the Marczewski map U : X → P(N) is a discrete limit of continuous
functions Ψ .
n
Clearly, every discrete limit is a quasi-normal limit. The proof of [10,
Theorem 4.8] actually establishes the following.
Lemma 12. Assume that P is a property of topological spaces, which is
preserved by taking closed subsets, continuous images and countable unions.
IfX hastheproperty P andΨ : X → Y isaquasi-normallimitofcontinuous
functions into a metric space Y, then Ψ[X] has the property P.
Proof. Let Ψ , n ∈ N, be continuous functions as in the premise of the
n
lemma,andletǫ ,n ∈ N,beasinthedefinitionofquasi-normalconvergence.
n
For each k,
X ={x ∈ X :(∀n,m ≥ k) d(Ψ (x),Ψ (x)) ≤ ǫ +ǫ }
k n m n m
HUREWICZ AND SHEAF AMALGAMATIONS 9
is a closed subset of X, and the functions Ψ converge to Ψ uniformly on
n
X . Thus, Ψ is continuous on X , and therefore Ψ[X ] has the property P.
k k k
Now, X = X , and therefore Ψ[X] = Ψ[X ] has the property
k k k k
P. (cid:3)
S S
It follows that for each countable family U of open subsets of X, U[X]
satisfies U (O,Γ).
fin
Let F,B denote the families of all countable closed and all countable
Borel covers of X, respectively. Similarly, let F ,B denote the families of
Γ Γ
all countable closed point-cofinite covers of X and all Borel point-cofinite
covers of X. Following isastrikingresultof Bukovsky´, Rec law, andRepicky´
[10]. In their terminology, it tells that the family of closed subsets of X is
weakly distributive if, and only if, the same holds for the family of Borel
subsets of X. In the language of selection principles, this result has the
following compact form.
Lemma 13 (Bukovsky´–Rec law–Repicky´ [10]). U (F,F )= U (B,B ).
fin Γ fin Γ
Proof. Assume that X satisfies U (F,F ). We first show that X is a σ
fin Γ
space [10, Theorem 5.2].
Assume that G = U where for each n, U ⊇ U are open subsets
n n n n+1
of X. Write, for each n,
T
U = Cn,
n m
m∈N
[
where for each m, Cn ⊆ Cn are closed subsets of X. We may assume
m m+1
that the closed cover {Cn ∪(X\U ): m ∈ N} of X has no finite subcover.5
m n
As X satisfies U (F,F ) and each given cover is monotone, there are m ,
fin Γ n
n ∈ N, such that {Cn ∪(X \U ) : n ∈ N} is a closed point-cofinite cover
mn n
of X. For each k define
∞
Z = Cn .
k mn
n=k
\
Then each Z is a closed subset of X, and G = Z is F . This shows
k k k σ
that X is a σ space.
Now, assume that U ∈ B, n∈ N. Then for eachSn, each element of U is
n n
F and can therefore be replaced by countably many closed sets. Applying
σ
U (F,F ) to the thus modified covers, we obtain a cover in F . For each
fin Γ Γ
n, extend each of the finitely many chosen elements of the nth cover to an
F set from the original cover U , to obtain an element of B chosen in
σ n Γ
accordance with the definition of U (B,B ).6 (cid:3)
fin Γ
Lemma 14. The following are equivalent:
(1) X satisfies U (B,B );
fin Γ
5If there are infinitely many n for which there is some mn with Cmnn = Un, then
G = T Cn is closed and we are done. Otherwise, we can ignore finitely many n and
n mn
assume that there are no n,m such that Cmn =Un contains G.
6This argument, in more general form, appears in [11, Theorem 2.1].
10 BOAZTSABANANDLYUBOMYRZDOMSKYY
(2) For each countable family U of open subsets of X, U[X] satisfies
U (O,Γ);
fin
(3) For each countable family C of closed subsets of X, C[X] satisfies
U (O,Γ).
fin
Proof. (2 ⇔ 3) Use the auto-homeomorphism of P(N) defined by mapping
a set to its complement.
(1 ⇒ 2) The Marczewski map U : X → P(N) is Borel. It is easy to
see that U (B,B ) is preserved by Borel images [39]. Thus, U[X] satisfies
fin Γ
U (B,B ), and in particular U (O,Γ).
fin Γ fin
(3 ⇒ 1) By Lemma 13, it suffices to show that X satisfies U (F,F ).
fin Γ
For each C = {C : n ∈ N} ∈ F which does not contain a finite subcover,
n
{ C : n ∈ N} ∈ F . Thus, U (F,F ) = U (F ,F ),7 and we prove
m≤n m Γ fin Γ fin Γ Γ
the latter property.
SLet C = {Cn : m ∈ N}, n ∈ N, be bijectively enumerated closed point-
n m
cofinite covers of X which do not contain finite subcovers. We may assume
that these covers are pairwise disjoint [35].
Let C = C , and consider the Marczewski map C : X → P(N × N)
n n
defined by
S C(x) = {(n,m) : x ∈ Cn}
m
forallx ∈ X. Foreach(n,m),O = {A⊆ N×N :(n,m) ∈ A}isanopen
(n,m)
subset of P(N×N), and for each n, U = {O :m ∈ N} is an open cover
n (n,m)
of C[X] that does not contain a finite subcover. As C[X] satisfies U (O,Γ),
fin
there are k , n ∈ N, such that { O : n ∈ N} is a point-cofinite
n m<kn (n,m)
cover of C[X]. Then { Cn : n ∈ N} is a point-cofinite cover of X (it
m<kn mS
is infinite because X does not appear there as an element).8 (cid:3)
S
ByLemma14,X satisfiesU (B,B ). Itremainstoobservethefollowing.
fin Γ
For the reader’s convenience, we reproduce the proof of the implication
needed in the present proof.
Lemma 15 (Bartoszyn´ski–Scheepers [4]). X satisfies U (B,B ) if, and
fin Γ
only if, each Borel image of X in NN is bounded.
Proof. (⇒) Assume that Y ⊆ NN is a Borel image of X. Then Y satisfies
U (B,B ). BytakingtheimageofY underthecontinuousmappingf(n)7→
fin Γ
f(1)+···+f(n) defined on NN , we may assume that all elements in Y are
nondecreasing.
We first consider the trivial case: There is an infinite I ⊆ N such that for
each n ∈ I, F ={f(n) :f ∈ Y} is finite. For each n, let m ∈ I be minimal
n
such that n≤ m, and define g(n) = maxF . Then Y is bounded by g.
m
Thus, assume that there is N such that for each n ≥ N, {f(n) : f ∈ Y}
is infinite. For all n,m, consider the open set Un = {f ∈ Y : f(n) ≤ m}.
m
7This statement holds in a more general form [22].
8The argument is standard: For each finite family of proper subsets of X, there is a
finitesubset of X not contained in any memberof thisfamily.