1 Introduction
Let
$\mathbb {N}$
be the set of natural numbers and
${\operatorname {P}(\mathbb {N})}$
be the power set of
$\mathbb {N}$
. We identify each set in
${\operatorname {P}(\mathbb {N})}$
with its characteristic function, an element of the Cantor cube
${\{0,1\}^{\mathbb {N}}}$
; in that way we introduce topology in
${\operatorname {P}(\mathbb {N})}$
. The Cantor space
${\operatorname {P}(\mathbb {N})}$
with the symmetric difference operation
$\oplus $
is a topological group; this operation coincides with the addition modulo
$2$
in
${\{0,1\}^{\mathbb {N}}}$
. A set
$X\subseteq {\operatorname {P}(\mathbb {N})}$
is null-additive in
${\operatorname {P}(\mathbb {N})}$
if for any null set
$Y\subseteq {\operatorname {P}(\mathbb {N})}$
the set
$X\oplus Y:=\{\,x\oplus y : x\in X, y\in Y\,\}$
is null. In an analogous way, define null-additive subsets of the real line with the addition
$+$
as a group operation. As we see in the forthcoming Theorem 2.2, it is relatively consistent with ZFC that null-additive subsets of
${\operatorname {P}(\mathbb {N})}$
are not preserved by homeomorphisms into
${\operatorname {P}(\mathbb {N})}$
. Subsets of the real line whose all continuous images into the real line are null-additive were considered by Galvin and Miller [Reference Galvin and Miller3]; to this end they used combinatorial covering properties.
By space we mean a Tychonoff topological space. A cover of a space is a family of proper subsets of the space whose union is the entire space. An open cover of a space is a cover whose members are open subsets of the space. A cover of a space is an
$\omega $
-cover if each finite subset of the space is contained in a set from the cover and it is a
$\unicode{x3b3} $
-cover if it is infinite and each point of the space belongs to all but finitely many sets from the cover. A space has the property
$\unicode{x3b3} $
if every open
$\omega $
-cover of the space contains a
$\unicode{x3b3} $
-cover. This property was introduced by Gerlits and Nagy in the context of local properties of functions spaces [Reference Gerlits and Nagy4]. They proved that a space X has the property
$\unicode{x3b3} $
if and only if the space
$\operatorname {C}_{\mathrm {p}}(X)$
of all continuous real-valued functions defined on X with the pointwise convergence topology is Fréchet–Urysohn, i.e., each point in the closure of a subset of
$\operatorname {C}_{\mathrm {p}}(X)$
is a limit of a sequence from the set [Reference Gerlits and Nagy4, Theorem 2]. Galvin and Miller observed that for a subset of the real line X with the property
$\unicode{x3b3} $
and a meager subset of the real line Y, the set
$X + Y:=\{\,x+y : x\in X, y\in Y\,\}$
is meager. They pointed out that they were unable to prove an analogous statement for null sets, and thus they introduced a formally stronger property than
$\unicode{x3b3} $
[Reference Galvin and Miller3, p. 152]. For a natural number n, an open cover of a space is an n-cover if each n-elements subset of the space is contained in a member of the cover. A space X has the property strongly
$\unicode{x3b3} $
if there is an increasing sequence
$f\in {\mathbb {N}^{\mathbb {N}}}$
such that for each sequence
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc $
where
$\mathcal {U}_n$
is an
$f(n)$
-cover of X, there are sets
$U_1\in \mathcal {U}_1,\, U_2\in \mathcal {U}_2, \dotsc $
such that the family
$\{U_n : n\in \mathbb {N}\}$
is a
$\unicode{x3b3} $
-cover of X.
Galvin and Miller proved that any subset of the real line with the property strongly
$\unicode{x3b3} $
is null-additive. The property strongly
$\unicode{x3b3} $
is preserved by continuous mappings, and thus any continuous image of a set with the property strongly
$\unicode{x3b3} $
, into the real line, is null-additive [Reference Galvin and Miller3, Theorem 7]. Under Martin Axiom, Galvin and Miller constructed a subset of
${\operatorname {P}(\mathbb {N})}$
of cardinality continuum with the property strong-
$\unicode{x3b3} $
[Reference Galvin and Miller3, Theorem 8]. In 1996, under some set-theoretic assumption, Bartoszyński and Recław [Reference Bartoszyński and Recław2] constructed a subset of
${\operatorname {P}(\mathbb {N})}$
of cardinality continuum with the property
$\unicode{x3b3} $
that is not null-additive (in particular, they separated the properties
$\unicode{x3b3} $
and strongly
$\unicode{x3b3} $
).
Assuming only an equality between some cardinal characteristics of the continuum, we construct a subset of
${\operatorname {P}(\mathbb {N})}$
of cardinality continuum with the property
$\unicode{x3b3} $
whose all continuous images into
${\operatorname {P}(\mathbb {N})}$
are null-additive and it contains a homeomorphic copy of a set that is not null-additive. We also weaken a set-theoretic assumption in the result of Bartoszyński and Recław. In both the cases we use combinatorial methods of construction of subsets of
${\operatorname {P}(\mathbb {N})}$
with the property
$\unicode{x3b3} $
invented by Tsaban ([Reference Orenshtein and Tsaban7, Theorem 3.6], [Reference Osipov, Szewczak and Tsaban8, Theorem 6]) and developed by Włudecka and the first named author [Reference Szewczak and Włudecka13]. We also use a combinatorial covering characterization of null-additive subsets of
${\operatorname {P}(\mathbb {N})}$
given by Zindulka [Reference Zindulka17].
2 Null-additive sets with the property
$\unicode{x3b3} $
Let
${[\mathbb {N}]^{\infty }}$
be the family of infinite subsets of
$\mathbb {N}$
. Each set in
${[\mathbb {N}]^{\tiny \infty }}$
we identify with an increasing function from
${\mathbb {N}^{\mathbb {N}}}$
. Depending on the interpretation, points of
${[\mathbb {N}]^{\tiny \infty }}$
are referred to as sets or functions. For natural numbers
$n,m$
with
$n<m$
, define
$[n,m):=\{\,i\in \mathbb {N} : n\leq i<m\,\}$
. For sets a and b, we write
$a\subseteq ^* b$
, if the set
$a\setminus b$
is finite. A pseudointersection of a family of infinite sets is an infinite set a with
$a\subseteq ^* b$
for all sets b in the family. A family of infinite sets is centered if the finite intersections of its elements are infinite. Let
$\mathfrak {p}$
be the minimal cardinality of a subfamily of
${[\mathbb {N}]^{\tiny \infty }}$
that is centered and has no pseudointersection. Let
$\mathrm {Fin}$
be the family of finite subsets of
$\mathbb {N}$
. The following notion plays a crucial role in our constructions.
Definition 2.1. (Szewczak and Włudecka [Reference Szewczak and Włudecka13])
A set
$X\subseteq {[\mathbb {N}]^{\tiny \infty }}$
with
$\left |X\right |\geq \mathfrak {p}$
is a
$\mathfrak {p}$
-generalized tower if for each function
$a\in {[\mathbb {N}]^{\tiny \infty }}$
, there are sets
$b\in {[\mathbb {N}]^{\tiny \infty }}$
and
$S\subseteq X$
with
$\left |S\right |<\mathfrak {p}$
such that
$$ \begin{align*} x\cap \bigcup_{n\in b}[a(n),a(n+1))\in\mathrm{Fin} \end{align*} $$
for all sets
$x\in X\setminus S$
.
For functions
$f,g\in {[\mathbb {N}]^{\tiny \infty }}$
, we write
$f\le ^* g$
, if the set
$\{\,n : f(n)>g(n)\,\}$
is finite. A subset of
${[\mathbb {N}]^{\tiny \infty }}$
is unbounded if for any function
$g\in {[\mathbb {N}]^{\tiny \infty }}$
, there is a function f in the set with
$f\not \le ^* g$
. Let
$\mathfrak {b}$
be the minimal cardinality of an unbounded subset of
${[\mathbb {N}]^{\tiny \infty }}$
. The existence of a
$\mathfrak {p}$
-generalized tower in
${[\mathbb {N}]^{\tiny \infty }}$
is independent of ZFC, i.e., it is equivalent to the equality
$\mathfrak {p}=\mathfrak {b}$
[Reference Orenshtein and Tsaban7, Lemma 3.3]. Let
$\operatorname {non}(\mathcal {N}_{\mathrm {add}})$
be the minimal cardinality of a subset of
${\operatorname {P}(\mathbb {N})}$
that is not null-additive.
Theorem 2.2. Assume that
$\mathfrak {p}=\operatorname {non}(\mathcal {N}_{\mathrm {add}})=\mathfrak {c}$
. There is a set
$X\subseteq {[\mathbb {N}]^{\tiny \infty }}$
such that
-
(1) The set X is a
$\mathfrak {p}$
-generalized tower. -
(2) The set
$X\cup \mathrm {Fin}$
has the property
$\unicode{x3b3} $
. -
(3) All continuous images of the set
$X\cup \mathrm {Fin}$
into
${\operatorname {P}(\mathbb {N})}$
are null-additive. -
(4) The set X is homeomorphic to a subset of
${\operatorname {P}(\mathbb {N})}$
that is not null-additive.
We need the following notions and auxiliary results.
For a class
$\mathcal {A}$
of covers of spaces and a space X, let
$\mathcal {A}(X)$
be the family of all covers of X from the class
$\mathcal {A}$
. Let
$\mathcal {A}_1, \mathcal {A}_2, \dotsc $
and
$\mathcal {B}$
be classes of covers of spaces. A space X satisfies
$$ \begin{align*} \mathsf{S}_1(\{\mathcal{A}_n\}_{n\in\mathbb{N}}, \mathcal{B}) \end{align*} $$
if for each sequence
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc $
with
$\mathcal {U}_n\in \mathcal {A}_n(X)$
for each n, there are sets
$U_1\in \mathcal {U}_1, U_2\in \mathcal {U}_2, \dotsc $
such that
$\{\,U_n : n\in \mathbb {N}\,\}\in \mathcal {B}(X)$
. Let
$\mathrm {O}_n$
be the class of all open n-covers of spaces for each n and
$\Gamma $
be the class of all open
$\unicode{x3b3} $
-covers of spaces. Using the above notion, the strongly
$\unicode{x3b3} $
property is the property
$\mathsf {S}_1(\{\mathrm {O}_n\}_{n\in \mathbb {N}}),\Gamma )$
. Let d be a metric in
${\operatorname {P}(\mathbb {N})}$
that coincides with the standard metric in
${\{0,1\}^{\mathbb {N}}}$
, that is, for points
$x,y\in {\operatorname {P}(\mathbb {N})}$
, let
$$ \begin{align*} d(x,y):= \begin{cases} 2^{-\min((x\cup y)\setminus (x\cap y))}, &\text{ if }x\neq y,\\ 0, &\text{ if }x=y. \end{cases} \end{align*} $$
Let
$\epsilon $
be a positive real number. For a point
$x\in {\operatorname {P}(\mathbb {N})}$
, let
$B(x,\epsilon ):=\{\,y\in {\operatorname {P}(\mathbb {N})} : d(x,y)<\epsilon \,\}$
and for a set
$F\subseteq {\operatorname {P}(\mathbb {N})}$
, let
$B(F,\epsilon ):=\bigcup \{\,B(x,\epsilon ) : x\in F\,\}$
. Fix a natural number n. An open cover of a subspace of
${\operatorname {P}(\mathbb {N})}$
is a uniform n-cover if there is a positive real number
$\epsilon $
such that for each n-elements subset F of the space, the set
$B(F,\epsilon )$
is contained in a member of the cover. Let
$\mathrm {O}_n^{\mathsf {unif}}$
be the family of all uniform n-covers of subspaces of
${\operatorname {P}(\mathbb {N})}$
. In 1995, Shelah characterized null-additive subsets of
${\operatorname {P}(\mathbb {N})}$
[Reference Shelah10]. Using the Shelah result, Zindulka characterized null-additive subsets of
${\operatorname {P}(\mathbb {N})}$
, in terms of combinatorial covering properties.
Theorem 2.3 (Zindulka [Reference Zindulka17])
A subset of
${\operatorname {P}(\mathbb {N})}$
is null-additive if and only if it satisfies
$\mathsf {S}_1(\{\mathrm {O}_n^{\mathsf {unif}}\}_{n\in \mathbb {N}},\Gamma )$
.
For a set X and a natural number n, let
$\operatorname {P}_n(X)$
be the family of all n-elements subsets of X. By a similar argument as in the proof of a result of Tsaban [Reference Tsaban14, Theorem 2.1], we have the following result.
Lemma 2.4. For a set
$X\subseteq {\operatorname {P}(\mathbb {N})}$
the following assertions are equivalent
$:$
-
(1) The set X satisfies
$\mathsf {S}_1(\{\mathrm {O}_n^{\mathsf {unif}}\}_{n\in \mathbb {N}},\Gamma )$
. -
(2) There is a function
$f\in {[\mathbb {N}]^{\tiny \infty }}$
such that the set X satisfies
$\mathsf {S}_1(\{\mathrm {O}_{f(n)}^{\mathsf {unif}}\}_{n\in \mathbb {N}},\Gamma )$
. -
(3) For each function
$f\in {[\mathbb {N}]^{\tiny \infty }}$
, the set X satisfies
$\mathsf {S}_1(\{\mathrm {O}_{f(n)}^{\mathsf {unif}}\}_{n\in \mathbb {N}},\Gamma )$
.
Proposition 2.5. Let
$X\subseteq {\operatorname {P}(\mathbb {N})}$
be a set and assume that there is a function
$f\in {[\mathbb {N}]^{\tiny \infty }}$
such that for each sequence
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc $
with
$\mathcal {U}_n\in \mathrm {O}_{f(n)}(X)$
for each n, there are sets
$U_1\in \mathcal {U}_1, U_2\in \mathcal {U}_2, \dotsc $
and a set
$X'\subseteq X$
with
$\left |X'\right |<\operatorname {non}(\mathcal {N}_{\mathrm {add}})$
such that
$\{\,U_n : n\in \mathbb {N}\,\}\in \Gamma (X\setminus X')$
. Then all continuous images of the set X into
${\operatorname {P}(\mathbb {N})}$
are null-additive.
Proof Let
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc $
be a sequence such that
$\mathcal {U}_n\in \mathrm {O}^{\mathsf {unif}}_{2f(n)}(X)$
for each n. Fix a natural number n. We may assume that there is a positive real number
$\epsilon _n$
such that
$$ \begin{align*} \mathcal{U}_n=\{\,B(A,\epsilon_n) : A\in\operatorname{P}_{2f(n)}(X)\,\}. \end{align*} $$
We have
$\mathcal {U}^{\prime }_n:=\{\,B(F,\epsilon _n) : F\in \operatorname {P}_{f(n)}(X)\,\}\in \mathrm {O}^{\mathsf {unif}}_{f(n)}(X)$
. By the assumption, there are sets
$F_1\in \operatorname {P}_{f(1)}(X), F_2\in \operatorname {P}_{f(2)}(X), \dotsc $
and a set
$X'\subseteq X$
with
$\left |X'\right |<\operatorname {non}(\mathcal {N}_{\mathrm {add}})$
such that
$$ \begin{align*} \{\,B(F_n,\epsilon_n) : n\in\mathbb{N}\,\}\in\Gamma(X\setminus X'). \end{align*} $$
Since
$\left |X'\right |<\operatorname {non}(\mathcal {N}_{\mathrm {add}})$
, there are sets
$F_1'\in \operatorname {P}_{f(1)}(X'), F_2'\in \operatorname {P}_{f(2)}(X'), \dotsc $
such that
$$ \begin{align*} \{\,B(F_n',\epsilon_n) : n\in\mathbb{N}\,\}\in\Gamma(X'). \end{align*} $$
For each natural number n, there is a set
$A_n\in \operatorname {P}_{2f(n)}(X)$
such that
$F_n\cup F_n'\subseteq A_n$
and
$$ \begin{align*} B(F_n,\epsilon_n)\cup B(F_n',\epsilon_n)\subseteq B(A_n,\epsilon_n)\in\mathcal{U}_n. \end{align*} $$
We have
$$ \begin{align*} \{\,B(A_n,\epsilon_n) : n\in\mathbb{N}\,\}\in \Gamma(X). \end{align*} $$
By Theorem 2.3, we have the following result.
Proposition 2.6. Null-additivity in
${\operatorname {P}(\mathbb {N})}$
is preserved by uniformly continuous functions.
Lemma 2.7. Let X be a space and
$n,k$
be natural numbers. If
$\mathcal {U}\in \mathrm {O}_{n+k}(X)$
, then each n-elements subset of X is contained in at least k pairwise different sets from
$\mathcal {U}$
.
For the remaining part of this section, let
$f\in {[\mathbb {N}]^{\tiny \infty }}$
be a sequence such that
$f(n+1)>2^{f(n)+2n}+n$
for all natural numbers n. For a set
$d\in {[\mathbb {N}]^{\tiny \infty }}$
, let
${[d]^{\tiny \infty }}$
be the family of all infinite subsets of d.
Lemma 2.8. Let
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc $
be a sequence of families of open sets in
${\operatorname {P}(\mathbb {N})}$
such that
$\mathcal {U}_n\in \mathrm {O}_{f(n)}(\mathrm {Fin})$
for all natural numbers n and
$d\in {[\mathbb {N}]^{\tiny \infty }}$
. There are an element
$x\in {[d]^{\tiny \infty }}$
and pairwise different sets
$U_1\in \mathcal {U}_1, U_2\in \mathcal {U}_2, \dotsc $
such that for each natural number n and each set
$y\in {\operatorname {P}(\mathbb {N})}:$
$$ \begin{align*} \text{If }y\cap [f(n),x(n+1))\subseteq \{x(1),x(1)+1,\dotsc,x(n), x(n)+1\},\text{ then }y\in U_{n+1}. \end{align*} $$
In particular,
$$ \begin{align*} \{\,U_n : n\in\mathbb{N}\,\}\in\Gamma(\{\,y\in{\operatorname{P}(\mathbb{N})} : y\subseteq^* x\cup(x+1)\,\}). \end{align*} $$
Proof Let
$x(1):=a(1)$
and
$U_1\in \mathcal {U}_1$
. Fix a natural number n and assume that natural numbers
$x(1),\dotsc ,x(n)$
and sets
$U_1\in \mathcal {U}_1,\dotsc ,U_n\in \mathcal {U}_n$
have already been defined. We have
$$ \begin{align*} \left|\operatorname{P}([1,f(n))\cup \{x(1),x(1)+1,\dotsc,x(n), x(n)+1\})\right|\leq 2^{f(n)+2n} \end{align*} $$
and
$\mathcal {U}_{n+1}\in \mathrm {O}_{f(n+1)}(\mathrm {Fin})$
. By Lemma 2.7 there is a set
$U_{n+1}\in \mathcal {U}_{n+1}\setminus \{U_1,\dotsc , U_n\}$
such that
$$ \begin{align*} \operatorname{P}([1,f(n))\cup \{x(1),x(1)+1,\dotsc,x(n), x(n)+1\})\subseteq U_{n+1}. \end{align*} $$
For each set
$$ \begin{align*} s\in \operatorname{P}([1,f(n))\cup \{x(1),x(1)+1,\dotsc,x(n), x(n)+1\}), \end{align*} $$
there is a natural number
$m_s\in a$
with
$m_s>\max \{f(n),x(n)+1\}$
such that for each set
$y\in {\operatorname {P}(\mathbb {N})}$
:
$$ \begin{align*} \text{If }y\cap [1,m_s)=s,\text{ then }y\in U_{n+1}. \end{align*} $$
Define
$$ \begin{align*} x(n+1):=\max\{\,m_s : s\in \operatorname{P}([1,f(n))\cup \{x(1),x(1)+1,\dotsc,x(n), x(n)+1\})\,\}. \end{align*} $$
Fix a natural number n. Let
$y\in {\operatorname {P}(\mathbb {N})}$
be a set such that
$$ \begin{align*} y\cap [f(n), x(n+1))\subseteq \{x(1),x(1)+1,\dotsc,x(n), x(n)+1\}. \end{align*} $$
Then the set
$s:=y\cap [1, x(n+1))$
belongs to
$\operatorname {P}([1,f(n))\cup \{x(1),x(1)+1,\dotsc ,x(n), x(n)+1\})$
, and
$$ \begin{align*} y\cap [1, m_s)=y\cap[1,x(n+1))=s. \end{align*} $$
Thus,
$y\in U_{n+1}$
.
Lemma 2.9. (Folklore [Reference Tsaban15, Lemma 2.13])
A set
$X\subseteq {[\mathbb {N}]^{\tiny \infty }}$
is unbounded if and only if for each function
$a\in {[\mathbb {N}]^{\tiny \infty }}$
, there are a set
$b\in {[\mathbb {N}]^{\tiny \infty }}$
and an element
$x\in X$
such that
$$ \begin{align*} x\cap \bigcup_{n\in b}[a(n),a(n+1))=\emptyset, \end{align*} $$
for all
$x\in X$
.
For elements
$b\in {[\mathbb {N}]^{\tiny \infty }}$
and
$c\in {\{0,1\}^{\mathbb {N}}}$
, let
$2b$
,
$b+1$
, and
$b+c$
be elements in
${[\mathbb {N}]^{\tiny \infty }}$
such that
$(2b)(n):=2b(n)$
,
$(b+1)(n):=b(n)+1$
, and
$(b+c)(n):=b(n)+c(n)$
for all natural numbers n.
Proof of Theorem 2.2 Construction of a set X: Let
$\{\,(\mathcal {U}^{(\alpha )}_1,\, \mathcal {U}^{(\alpha )}_2, \dotsc ) : \alpha <\mathfrak {c}\,\}$
be an enumeration of all sequences of families of open sets in
${\operatorname {P}(\mathbb {N})}$
such that
$\mathcal {U}^{(\alpha )}_n\in \mathrm {O}_{f(n)}(\mathrm {Fin})$
for all ordinal numbers
$\alpha <\mathfrak {c}$
and natural numbers n. Let
${[\mathbb {N}]^{\tiny \infty }}=\{\,d_\alpha : \alpha <\mathfrak {c}\,\}$
.
Apply Lemma 2.8 to the sequence
$\mathcal {U}^{(0)}_1,\, \mathcal {U}^{(0)}_2, \dotsc $
and to a set
$d\in {[2\mathbb {N}]^{\tiny \infty }}$
with
$d_0\le ^* d$
. Then there are sets
$U^{(0)}_1\in \mathcal {U}^{(0)}_1,U^{(0)}_2\in \mathcal {U}^{(0)}_2,\dotsc $
and an element
$x_0\in {[2\mathbb {N}]^{\tiny \infty }}$
such that
$$ \begin{align*} \{\,U_n^{(0)} : n\in\mathbb{N}\,\}\in\Gamma(\{\,y\in{\operatorname{P}(\mathbb{N})} : y\subseteq^* x_0\cup(x_0+1)\,\}). \end{align*} $$
Fix an ordinal number
$\alpha <\mathfrak {c}$
. Assume that elements
$x_\beta \in {[2\mathbb {N}]^{\tiny \infty }}$
with
$d_\beta \le ^* x_\beta $
and sets
$U^{(\beta )}_1\in \mathcal {U}^{(\beta )}_1, U^{(\beta )}_2\in \mathcal {U}^{(\beta )}_2,\dotsc $
with
$$ \begin{align*} \{\,U^{(\beta)}_n : n\in\mathbb{N}\,\}\in\Gamma(\{\,y\in{\operatorname{P}(\mathbb{N})} : y\subseteq^*x_\beta\cup(x_\beta+1)\,\} \end{align*} $$
have already been defined for all ordinal numbers
$\beta <\alpha $
such that for
$\beta ,\beta '<\alpha $
if
$\beta <\beta '$
, then
$x_\beta {}^{*}{\supseteq }\ x_{\beta '}$
. Let
$d\in {[2\mathbb {N}]^{\tiny \infty }}$
be a pseudointersection of the family
$\{\,x_\beta : \beta <\alpha \,\}$
with
$d_\alpha \le ^* d$
. Apply Lemma 2.8 to the sequence
$\mathcal {U}^{(\alpha )}_1, \mathcal {U}^{(\alpha )}_2, \dotsc $
and to the element d. Then there are an element
$x_\alpha \in {[d]^{\tiny \infty }}$
and sets
$U^{(\alpha )}_1\in \mathcal {U}^{(\alpha )}_1, U^{(\alpha )}_2\in \mathcal {U}^{(\alpha )}_2,\dotsc $
such that
$$ \begin{align} \{\,U_n^\alpha : n\in\mathbb{N}\,\}\in\Gamma(\{\,y\in{\operatorname{P}(\mathbb{N})} : y\subseteq^* x_\alpha\cup(x_\alpha+1)\,\}). \end{align} $$
Let
${\{0,1\}^{\mathbb {N}}}=\{\,c_\alpha : \alpha <\mathfrak {c}\,\}$
. Define
$$ \begin{align*} X:=\{\,x_\alpha+c_\alpha : \alpha<\mathfrak{c}\,\}. \end{align*} $$
By the construction, for all ordinal numbers
$\alpha ,\beta $
with
$\alpha \leq \beta <\mathfrak {c}$
, we have
$$ \begin{align} x_\beta+c_\beta\subseteq x_\beta\cup (x_\beta+1)\subseteq^* x_\alpha\cup(x_\alpha+1). \end{align} $$
(1) Let
$a\in {[\mathbb {N}]^{\tiny \infty }}$
. Since the set
$\{\,x_\alpha : \alpha <\mathfrak {c}\,\}$
is unbounded, the set
$\{\,x_\alpha \cup (x_\alpha +1) : \alpha <\mathfrak {c}\,\}$
is unbounded, too. By Lemma 2.9, there are an ordinal number
$\alpha <\mathfrak {c}$
and a set
$b\in {[\mathbb {N}]^{\tiny \infty }}$
such that
$$ \begin{align*} (x_\alpha\cup (x_\alpha+1))\cap\bigcup_{n\in b}[a(n),a(n+1))=\emptyset. \end{align*} $$
By (2.9.2), we have
$$ \begin{align*} (x_\beta+c_\beta) \cap\bigcup_{n\in b}[a(n),a(n+1))\subseteq^* (x_\alpha\cup (x_\alpha+1))\cap\bigcup_{n\in b}[a(n),a(n+1))=\emptyset \end{align*} $$
for all ordinal numbers
$\beta $
with
$\alpha \leq \beta <\mathfrak {c}$
. Thus, the set X is a
$\mathfrak {p}$
-generalized tower.
(2) By (1), the set X is a
$\mathfrak {p}$
-generalized tower, and thus the set
$X\cup \mathrm {Fin}$
has the property
$\unicode{x3b3} $
[Reference Szewczak and Włudecka13, Theorem 4.1(1)].
(3) The set
$X\cup \mathrm {Fin}$
satisfies the property from Proposition 2.5: Let
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc $
be a sequence of open families in
${\operatorname {P}(\mathbb {N})}$
such that
$\mathcal {U}_n\in \mathrm {O}_{f(n)}(X\cup \mathrm {Fin})$
for all natural numbers n. There is an ordinal number
$\alpha <\mathfrak {c}$
such that this sequence is equal to the sequence
$\mathcal {U}^{(\alpha )}_1,\, \mathcal {U}^{(\alpha )}_2, \dotsc $
. By (2.9.1) and (2.9.2), we have
$$ \begin{align*} \{\,U^{(\alpha)}_n : n\in\mathbb{N}\,\}\in\Gamma(\{\,x_\beta+c_\beta : \alpha\leq\beta<\mathfrak{c}\,\}). \end{align*} $$
(4) In our proof we use Rothberger’s trick [Reference Miller6, Theorem 9.4]. The map
$\Phi \colon {\operatorname {P}(\mathbb {N})}\to {\{0,1\}^{\mathbb {N}}}$
unifying each subset of
$\mathbb {N}$
with its characteristic function is a homeomorphism and an isometry with respect to the standard metric in
${\{0,1\}^{\mathbb {N}}}$
and the metric d on
${\operatorname {P}(\mathbb {N})}$
defined in Section 2. The set
$$ \begin{align*} G:={\bigl\{\,((\Phi(x))(1),\, x(1)_{\bmod 2},(\Phi(x))(2),\, x(2)_{\bmod 2},\dotsc) : x\in X\,\bigr\}} \end{align*} $$
is a homeomorphic copy of the set X. Since the projection
$\pi \colon G\to \{0,1\}^{2\mathbb {N}}$
is onto and uniformly continuous, the map
$$ \begin{align*} \Phi^{-1}\circ \pi \circ \Phi\colon \Phi^{-1}[G]\to {\operatorname{P}(\mathbb{N})} \end{align*} $$
is onto and uniformly continuous, too. By Proposition 2.6, the set
$\Phi ^{-1}[G]$
is not null additive. The set
$\Phi ^{-1}[G]$
is homeomorphic to G.
3 A
$\unicode{x3b3} $
-set that is not null-additive
Let
$\operatorname {non}(\mathcal {N})$
be the minimal cardinality of a subset of
${\operatorname {P}(\mathbb {N})}$
that is not null. A set
$\{\,x_\alpha : \alpha <\mathfrak {p}\,\}\subseteq {[\mathbb {N}]^{\tiny \infty }}$
is a
$\mathfrak {p}$
-unbounded tower if it is unbounded and for all ordinal numbers
$\alpha ,\beta <\mathfrak {p}$
with
$\alpha <\beta $
, we have
$x_{\alpha }{}^{*}{\supseteq }\ x_\beta $
. A
$\mathfrak {p}$
-unbounded tower exists if and only if
$\mathfrak {p}=\mathfrak {b}$
[Reference Orenshtein and Tsaban7, Lemma 3.3].
Theorem 3.1. Assume that
$\mathfrak {p}=\mathfrak {b}=\operatorname {non}(\mathcal {N})$
. There is a
$\mathfrak {p}$
-generalized tower
$X\subseteq {[\mathbb {N}]^{\tiny \infty }}$
that is not null-additive. In particular, the set
$X\cup \mathrm {Fin}$
is a nontrivial set with the property
$\unicode{x3b3} $
that is not null-additive.
Proof Let
$f\in {[\mathbb {N}]^{\tiny \infty }}$
be a function such that
$f(n):=\sum _{i=0}^{n}2^i$
for all natural numbers n. Define
$$ \begin{align*} Y_n:=\{\,x\in{\operatorname{P}(\mathbb{N})} : x\cap [f(n),f(n+1))=\emptyset\,\} \end{align*} $$
for all natural numbers n. Each set
$Y_n$
is clopen and has measure less or equal than
$2^{-(n+1)}$
, and thus the set
$Y:=\bigcap _m\bigcup _{n>m}Y_n$
is null. Let
$T=\{\,t_\alpha : \alpha <\mathfrak {p}\,\}$
be a
$\mathfrak {p}$
-unbounded tower in
${[\mathbb {N}]^{\tiny \infty }}$
and
$Z=\{\,z_\alpha : \alpha <\mathfrak {p}\,\}$
be a nonnull set in
${\operatorname {P}(\mathbb {N})}$
.
Define
$$ \begin{align*} x_\alpha:=z_\alpha\cap \bigcup_{n\in\mathbb{N}}[f(t_\alpha(2n)),f(t_\alpha(2n)+1) \cup\{\,f(t_\alpha(2n+1)) : n\in\mathbb{N}\,\} \end{align*} $$
for all ordinal numbers
$\alpha <\mathfrak {p}$
and
$X:=\{\,x_\alpha : \alpha <\mathfrak {p}\,\}$
.
The set X is a
$\mathfrak {p}$
-generalized tower: For each ordinal number
$\alpha $
with
$\alpha <\mathfrak {p}$
, we have
$$ \begin{align*} \{\,f(t_\alpha(2n+1)) : n\in\mathbb{N}\,\}\subseteq x_\alpha, \end{align*} $$
and thus
$X\subseteq {[\mathbb {N}]^{\tiny \infty }}$
. Let
$a\in {[\mathbb {N}]^{\tiny \infty }}$
. There is a set
$c\in {[\mathbb {N}]^{\tiny \infty }}$
such that
$$ \begin{align*} \left|[f(c(k)),f(c(k+1)))\cap a\right|\geq 2 \end{align*} $$
for all natural numbers k. By Lemma 2.9, there are a set
$d\in {[\mathbb {N}]^{\tiny \infty }}$
and an ordinal number
$\alpha <\mathfrak {p}$
such that
$$ \begin{align} t_\alpha\cap\bigcup_{k\in d}[c(k),c(k+1))=\emptyset. \end{align} $$
Then there is a set
$b\in {[\mathbb {N}]^{\tiny \infty }}$
such that
$$ \begin{align*} \bigcup_{n\in b}[a(n),a(n+1))\subseteq \bigcup_{k\in d}[f(c(k)),f(c(k+1))). \end{align*} $$
Fix an ordinal number
$\beta $
with
$\alpha \leq \beta <\mathfrak {p}$
. We have
$$ \begin{gather*} x_\beta\cap \bigcup_{n\in b}[a(n),a(n+1))\subseteq \bigcup_{n\in t_\beta}[f(n),f(n+1))\cap \bigcup_{n\in b}[a(n),a(n+1))\\\subseteq^* \bigcup_{n\in t_\alpha}[f(n),f(n+1))\cap \bigcup_{k\in d}[f(c(k)),f(c(k+1))). \end{gather*} $$
By (3.1.1), the latter intersection is empty.
We have
$Z\subseteq X\oplus Y$
: Fix an ordinal number
$\alpha <\mathfrak {p}$
. Since
$$ \begin{align*} x_\alpha\cap \bigcup_{n\in\mathbb{N}}[f(t_\alpha(2n)),f(t_\alpha(2n)+1)= z_\alpha\cap \bigcup_{n\in\mathbb{N}}[f(t_\alpha(2n)),f(t_\alpha(2n)+1), \end{align*} $$
we have
$$ \begin{align*} (x_\alpha\oplus z_\alpha)\cap \bigcup_{n\in\mathbb{N}}[f(t_\alpha(2n)),f(t_\alpha(2n)+1)=\emptyset , \end{align*} $$
and thus
$x_\alpha \oplus z_\alpha \in Y$
. Then there is an element
$y_\alpha \in Y$
such that
$x_\alpha \oplus z_\alpha =y_\alpha $
. Thus,
$z_\alpha =x_\alpha \oplus y_\alpha $
.
Since X is a
$\mathfrak {p}$
-generalized tower, the set
$X\cup \mathrm {Fin}$
has the property
$\unicode{x3b3} $
[Reference Szewczak and Włudecka13, Theorem 4.1(1)]. Since
$Z\subseteq (X\cup \mathrm {Fin})\oplus Y$
, the set
$X\cup \mathrm {Fin}$
is not null-additive.
Remark 3.2. The assumption of Theorem 3.1 is valid assuming the Continuum Hypothesis or Martin Axiom and in the following model. Let
$\mathbb {P}_{\aleph _1}$
be an
$\aleph _1$
-iteration with finite support of a measure algebra and G be a generic filter in
$\mathbb {P}_{\aleph _1}$
. Let M be a model of ZFC and
$\mathfrak {c}=\aleph _2$
. In the model
$M[G]$
, the assumption
$\mathfrak {p}=\mathfrak {b}=\operatorname {non}(\mathcal {N})$
from Theorem 3.1 is true: Let
$\operatorname {non}(\mathcal {M})$
be the minimal cardinality of a subset of
${\operatorname {P}(\mathbb {N})}$
that is not meager. Adding
$\aleph _1$
Cohen reals, we have
$\operatorname {non}(\mathcal {M})=\aleph _1$
and adding
$\aleph _1$
Solovay reals, we have
$\operatorname {non}(\mathcal {N})=\aleph _1$
. Thus,
$\mathfrak {p}=\aleph _1$
. Since
$\operatorname {non}(\mathcal {M})=\aleph _1$
, we have
$\mathfrak {b}=\aleph _1$
.
Theorem 3.3. Assume that
$\mathfrak {p}=\mathfrak {b}=\operatorname {non}(\mathcal {N})$
. There is a nontrivial subset of the real line with the property
$\unicode{x3b3} $
that is not null-additive.
Proof We use notions from the paper of the second named author [Reference Weiss16, Corollary 11]. Let
$X\subseteq {[\mathbb {N}]^{\tiny \infty }}$
be a set from Theorem 3.1. Let
$p\colon {\operatorname {P}(\mathbb {N})}\to A$
be a homeomorphism. Since the map
$f\circ g^{-1}\circ p$
is continuous, the image Y of the set
$X\cup \mathrm {Fin}$
under the map
$f\circ g^{-1}\circ p$
is a subset of the real line with the property
$\unicode{x3b3} $
. Suppose that the set Y is null-additive. Then the set
$g\circ f^{-1}[Y]$
is null-additive [Reference Weiss16, Theorem 12]. Since the map
$p^{-1}$
is uniformly continuous and the set
$X\cup \mathrm {Fin}$
is not null-additive, the set
$p[X]=(g\circ f^{-1}[Y])$
is not null-additive, too, a contradiction.
Theorem 3.4. It is consistent with ZFC that there is a
$\mathfrak {p}$
-unbounded tower in
${[\mathbb {N}]^{\tiny \infty }}$
and
$\operatorname {non}(\mathcal {N}_{\mathrm {add}})<\mathfrak {p}$
.
Proof The dual Borel conjecture is the statement that for any set
$X\subseteq {\operatorname {P}(\mathbb {N})}$
with cardinality
$\aleph _1$
, there is a null set
$N\subseteq {\operatorname {P}(\mathbb {N})}$
such that
$X\oplus N={\operatorname {P}(\mathbb {N})}$
. By the result of Judah and Shelah [Reference Bartoszyński and Judah1, Theorem 8.5.23] there is a model for ZFC satisfying Martin Axiom for
$\sigma $
-centered sets and the dual Borel conjecture. In that model, we have
$\mathfrak {p}=\aleph _2=\mathfrak {b}$
, and thus a
$\mathfrak {p}$
-unbounded tower exists. On the other hand, since the dual Borel conjecture holds, we have
$\operatorname {non}(\mathcal {N}_{\mathrm {add}})=\aleph _1$
.
4 Nonproductivity of Sierpiński-type sets
Let
$\mathrm {O}$
be the class of open covers of spaces. A space X satisfies Menger’s property
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
if for each sequence
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc \in \mathrm {O}(X)$
, there are finite sets
$\mathcal {F}_1\subseteq \mathcal {U}_1,\,\mathcal {F}_2\subseteq \mathcal {U}_2,\dotsc $
such that
$\bigcup _n\mathcal {F}_n\in \mathrm {O}(X)$
. A space X satisfies Hurewicz’s property
$\mathsf {U}_{\mathrm {fin}}(\mathrm {O},\Gamma )$
if for each sequence
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc \in \mathrm {O}(X)$
, there are finite sets
$\mathcal {F}_1\subseteq \mathcal {U}_1,\, \mathcal {F}_2\subseteq \mathcal {U}_2,\dotsc $
such that
$\{\,\bigcup \mathcal {F}_n : n\in \mathbb {N}\,\}\in \Gamma (X)$
. The property
$\mathsf {U}_{\mathrm {fin}}(\mathrm {O},\Gamma )$
implies
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
and it generalizes
$\sigma $
-compactness. An uncountable subset of
${\operatorname {P}(\mathbb {N})}$
is a Sierpiński set if its intersection with any null set is at most countable; its existence is independent of ZFC. Any Sierpiński set satisfies
$\mathsf {U}_{\mathrm {fin}}(\mathrm {O},\Gamma )$
but it is not
$\sigma $
-compact. Assuming that the Continuum Hypothesis holds, there are two Sierpiński sets whose product space does not even satisfy
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
[Reference Just, Miller, Scheepers and Szeptycki5, p. 250].
A category theoretic counterpart to a Sierpiński set is a Luzin set, i.e., an uncountable subset of
${\operatorname {P}(\mathbb {N})}$
whose intersection with any meager set is at most countable. Each Luzin set satisfies
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
but no
$\mathsf {U}_{\mathrm {fin}}(\mathrm {O},\Gamma )$
. A set of reals is a space homeomorphic to a subset of the real line. Let
$\mathbf{P}$
be a property of spaces. A space is productively
$\mathbf{P}$
if its product space with any space satisfying
$\mathbf{P}$
satisfies
$\mathbf{P}$
. Assuming
$\aleph _1=\operatorname {cf}(\mathfrak {d})$
, in the class of sets of reals, no Luzin set is productively
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
[Reference Szewczak and Tsaban11, Corollary 2.11]. There are open problems ([Reference Szewczak and Tsaban11, Problem 7.5], [Reference Szewczak and Tsaban12, Problem 5.5]), whether in the class of sets of reals (or in the class of general topological spaces) for any Sierpiński set X, there is a space Y satisfying
$\mathsf {U}_{\mathrm {fin}}(\mathrm {O},\Gamma )$
such that the product space
$X\times Y$
does not satisfy
$\mathsf {U}_{\mathrm {fin}}(\mathrm {O},\Gamma )$
, and what if we assume the Continuum Hypothesis? We consider an analogous problem with respect to combinatorial covering properties, stronger than
$\mathsf {U}_{\mathrm {fin}}(\mathrm {O},\Gamma )$
.
Let
$\mathcal {A}$
,
$\mathcal {B}$
be classes of covers of spaces. A space X satisfies
$\mathsf {S}_1(\mathcal {A},\mathcal {B})$
if for each sequence
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc \in \mathcal {A}(X)$
, there are sets
$U_1\in \mathcal {U}_1, U_2\in \mathcal {U}_2, \dotsc $
such that
$\{\,U_n : n\in \mathbb {N}\,\}\in \mathcal {B}(X)$
. Let
$\Omega $
be the class of all open
$\omega $
-covers of spaces. A Borel cover of a space is a cover whose members are Borel subsets of the space. Let
$\Gamma _{\mathrm {B}}$
and
$\Omega _{\mathrm {B}}$
be classes of all countable Borel
$\unicode{x3b3} $
-covers and countable Borel
$\omega $
-covers of spaces, respectively. The property
$\unicode{x3b3} $
, considered in the previous sections, is one of the classic properties in the selection principles theory, it is equivalent to the property
$\mathsf {S}_1(\Omega ,\Gamma )$
[Reference Gerlits and Nagy4, Theorem 2]. A set of reals is totally imperfect if it does not contain an uncountable compact set and it is perfectly meager if its intersection with a perfect set is meager in this perfect set. The following diagram presents relations between considered properties [Reference Just, Miller, Scheepers and Szeptycki5, Reference Scheepers9].
Let
$\operatorname {cov}(\mathcal {N})$
be the minimal cardinality of a family of null subsets of
${\operatorname {P}(\mathbb {N})}$
whose union is
${\operatorname {P}(\mathbb {N})}$
and
$\operatorname {cof}(\mathcal {N})$
be the minimal cardinality of a family of null subsets of
${\operatorname {P}(\mathbb {N})}$
such that any null subset of
${\operatorname {P}(\mathbb {N})}$
is contained in a set from the family. For an uncountable ordinal number
$\kappa $
, a set
$X\subseteq {\operatorname {P}(\mathbb {N})}$
is a
$\kappa $
-Sierpiński set if
$\left |X\right |\geq \kappa $
and for any null set
$Y\subseteq {\operatorname {P}(\mathbb {N})}$
, we have
$\left |X\cap Y\right |<\kappa $
. Every
$\mathfrak {b}$
-Sierpiński set satisfies
${\mathsf {S}_1(\Gamma _B,\Gamma _B)}$
([Reference Just, Miller, Scheepers and Szeptycki5, Theorem 2.9], [Reference Tsaban15, Theorem 2.4]).
-
(1) Assume that
$\operatorname {cov}(\mathcal {N})=\operatorname {cof}(\mathcal {N})=\mathfrak {b}$
. For every
$\mathfrak {b}$
-Sierpiński set S, there is a
$\mathfrak {b}$
-Sierpiński set
$S'$
such that the product space
$S\times S'$
does not satisfy
${\mathsf {S}_1(\Gamma ,\Gamma )}$
. -
(2) Assume that
$\operatorname {cov}(\mathcal {N})=\operatorname {cof}(\mathcal {N})=\mathfrak {d}=\mathfrak {c}$
and the cardinal number
$\mathfrak {c}$
is regular. For every
$\mathfrak {c}$
-Sierpiński set S, there is a
$\mathfrak {c}$
-Sierpiński set
$S'$
such that the product space
$S\times S'$
does not satisfy
${\mathsf {S}_1(\Gamma ,\mathrm {O})}$
.
In order to prove Theorem 4.1, we need the following lemma.
Lemma 4.2. Assume that
$\operatorname {cov}(\mathcal {N})=\operatorname {cof}(\mathcal {N})$
and the cardinal number
$\operatorname {cov}(\mathcal {N})$
is regular. For every
$\operatorname {cov}(\mathcal {N})$
-Sierpiński set S and every set Y of cardinality at most
$\operatorname {cov}(\mathcal {N})$
, there is a
$\operatorname {cov}(\mathcal {N})$
-Sierpiński set
$S'$
such that
$Y\subseteq S\oplus S'$
.
Proof Let
$\{\,N_\alpha : \alpha <\operatorname {cov}(\mathcal {N})\,\}$
be a cofinal family of null sets in
${\operatorname {P}(\mathbb {N})}$
and
$Y=\{\,y_\alpha : \alpha <\operatorname {cov}(\mathcal {N})\,\}\subseteq {\operatorname {P}(\mathbb {N})}$
. Fix an ordinal number
$\alpha <\operatorname {cov}(\mathcal {N})$
and assume that elements
$x_ \beta \in {\operatorname {P}(\mathbb {N})}$
have already been defined for all ordinal numbers
$\beta $
with
$\beta <\alpha $
. The set
$y_\alpha \oplus S$
is a
$\operatorname {cov}(\mathcal {N})$
-Sierpiński. Since the cardinal number
$\operatorname {cov}(\mathcal {N})$
is regular, the union
$\bigcup _{\beta <\alpha }N_\beta \cup \{\,x_\beta : \beta <\alpha \,\}$
cannot cover any
$\operatorname {cov}(\mathcal {N})$
-Sierpiński set. Then there is an element
$$ \begin{align*} x_\alpha\in (y_\alpha \oplus S)\setminus \bigcup_{\beta<\alpha}N_\beta\cup\{\,x_\beta : \beta<\alpha\,\}. \end{align*} $$
By the construction the set
$S':=\{\,x_\alpha : \alpha <\operatorname {cov}(\mathcal {N})\,\}$
is a
$\operatorname {cov}(\mathcal {N})$
-Sierpiński set and for each ordinal number
$\alpha <\operatorname {cov}(\mathcal {N})$
, there is an element
$s_\alpha \in S$
such that
$y_\alpha =x_\alpha \oplus s_\alpha $
. Thus,
$Y\subseteq S\oplus S'$
.
Let
$\operatorname {non}(\mathcal {M})$
be the minimal cardinality of a nonmeager subset of
${\operatorname {P}(\mathbb {N})}$
.
Proof of Theorem 4.1 (1) By the Cichoń diagram, we have
$\operatorname {cov}(\mathcal {N})\leq \operatorname {non}(\mathcal {M})\leq \operatorname {cof}(\mathcal {N})$
, and thus
$\operatorname {non}(\mathcal {M})=\mathfrak {b}$
. Let
$Y\subseteq {\operatorname {P}(\mathbb {N})}$
be a nonmeager set of cardinality
$\mathfrak {b}$
. By Lemma 4.2, there is a
$\mathfrak {b}$
-Sierpiński set
$S'$
such that
$Y\subseteq S\oplus S'$
. Every set with the property
${\mathsf {S}_1(\Gamma ,\Gamma )}$
is meager and the property
${\mathsf {S}_1(\Gamma ,\Gamma )}$
is preserved by continuous functions. Since
$S\oplus S'$
is a nonmeager continuous image of the product space
$S\times S'$
, the product
$S\times S'$
does not satisfy
${\mathsf {S}_1(\Gamma ,\Gamma )}$
.
(2) Every set satisfying
${\mathsf {S}_1(\Gamma ,\mathrm {O})}$
is totally imperfect. Let
$Y={[\mathbb {N}]^{\tiny \infty }}$
. Proceed analogously as in (1).
Remark 4.3. The assumptions of Theorem 4.1 are valid assuming the Continuum Hypothesis or Martin Axiom. In a model obtained by
$\aleph _2$
-iteration of Sacks forcing with countable supports, we have
$\operatorname {cov}(\mathcal {N})=\operatorname {non}(\mathcal {M})=\operatorname {cof}(\mathcal {N})=\mathfrak {b}=\aleph _1$
. In a model obtained by
$\aleph _2$
-iteration of amoeba forcing with finite supports, we have
$\operatorname {cov}(\mathcal {N})=\operatorname {cof}(\mathcal {N})=\mathfrak {d}=\mathfrak {c}$
.
It is not known whether, in the class of totally imperfect sets, the properties
${\mathsf {S}_1(\Gamma ,\mathrm {O})}$
and
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
are different. Theorem 4.1 can be useful to solve this problem.
Remark 4.4. Assume that the Continuum Hypothesis holds. If there is a Sierpiński set whose product space with any Sierpiński set satisfies
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
, then, in the class of totally imperfect sets, the properties
$\mathsf {S}_{\mathrm {fin}}(\mathrm {O},\mathrm {O})$
and
${\mathsf {S}_1(\Gamma ,\mathrm {O})}$
are different.
For a finite set
$F\subseteq {[\mathbb {N}]^{\tiny \infty }}$
and a set
$X\subseteq {[\mathbb {N}]^{\tiny \infty }}$
let
$\max [F]:=\{\,\max _{f\in F}f(n) : n\in \mathbb {N}\,\}$
, an element of
${[\mathbb {N}]^{\tiny \infty }}$
, and
$\operatorname {maxfin}[X]:=\{\,\max [F] : F\text { is a finite subset of }X\,\}$
. For elements
$f,g\in {[\mathbb {N}]^{\tiny \infty }}$
we write
$f\le ^{\infty } g$
if the set
$\{\,n : f(n)\leq g(n)\,\}$
is infinite.
Using similar ideas as in the proof that any Sierpiński set satisfies
${\mathsf {S}_1(\Gamma ,\Gamma )}$
[Reference Just, Miller, Scheepers and Szeptycki5, Theorems 2.9 and 2.10], we obtain the following results.
Proposition 4.5. Every
$\mathfrak {d}$
-Sierpiński set satisfies
${\mathsf {S}_1(\Gamma _{\mathrm {B}},\Omega _{\mathrm {B}})}$
.
Proof Let S be a
$\mathfrak {d}$
-Sierpiński set. There is a positive real number p such that the outer measure of S is equal to p. Let B be a Borel set containing S with
$\mu (B)=p$
. Let
$\mathcal {U}_1,\, \mathcal {U}_2, \dotsc \in \Gamma _{\mathrm {Bor}}(S)$
be a sequence of families of Borel subsets of the set B. Let
$\mathcal {U}_n=\{\,U^n_m : m\in \mathbb {N}\,\}$
for all natural numbers n, and we may assume that each such a family is increasing (if not consider the family
$\{\,\bigcap _{i\geq j}{U^n_i} : j\in \mathbb {N}\,\}$
, an increasing countable Borel cover of S). Fix a natural number k. There is a function
$f_k\in {\mathbb {N}^{\mathbb {N}}}$
such that
$\mu (U^n_{f_k(n)})\geq (1 - \frac {1}{2^{n+k}})p$
. For a set
$A_k:=\bigcap _{n}U^n_{f_k(n)}$
, we have
$\mu (A_k)\geq (1-\frac {1}{2^k})p$
. The set
$A=\bigcup _kA_k$
has measure p, and thus
$\left |S\setminus A\right |<\mathfrak {d}$
. For each element
$x\in S\setminus A$
, there is a function
$f_x\in {\mathbb {N}^{\mathbb {N}}}$
such that
$x\in \bigcap _nU^n_{f_x(n)}$
. There is a function
$g\in {\mathbb {N}^{\mathbb {N}}}$
such that
$$ \begin{align*} \operatorname{maxfin}[\{\,f_k : k\in\mathbb{N}\,\}\cup\{\,f_x : x\in S\setminus A\,\}]\le^{\infty} g. \end{align*} $$
We have
$\{\,U^n_{g(n)} : n\in \mathbb {N}\,\}\in \Omega (S)$
: Let F be a finite subset of S. There is a set
$a\in \mathrm {Fin}$
such that
$F\cap A\subseteq \bigcup _{k\in a}A_k$
. Let
$$ \begin{align*} f:=\operatorname{maxfin}[\{\,f_k : k\in a\,\}\cup\{\,f_x : x\in S\setminus A\,\}]. \end{align*} $$
Then
$F\subseteq U^n_{f(n)}$
for all but finitely many natural numbers n. Since
$f\le ^{\infty } g$
, and families
$\mathcal {U}_n$
are increasing, we have
$F\subseteq U^n_{g(n)}$
for infinitely many natural numbers n.
The following corollary is a straightforward consequence of Theorem 4.1.
-
(1) Assume that
$\operatorname {cov}(\mathcal {N})=\operatorname {cof}(\mathcal {N})=\mathfrak {b}$
. No
$\operatorname {cov}(\mathcal {N})$
-Sierpiński set is productively
${\mathsf {S}_1(\Gamma ,\Gamma )}$
. -
(2) Assume that
$\operatorname {cov}(\mathcal {N})=\operatorname {cof}(\mathcal {N})=\mathfrak {d}=\mathfrak {c}$
and the cardinal number
$\mathfrak {c}$
is regular. No
$\mathfrak {c}$
-Sierpiński set S is productively
${\mathsf {S}_1(\Gamma ,\Omega )}$
or productively
${\mathsf {S}_1(\Gamma ,\mathrm {O})}$
.
Acknowledgment
We would like to thank the referees for their work on refereeing this paper and their detailed comments and corrections.
