数理解析研究所講究録 1202 巻 2001 年 66-74

66
CARDINAL INVARIANTS RELATEDTO SEQUENTIAL SEPARABILITYMICHAEL HRU\v{S}\’AK AND JURIS STEPR\={A}NS
ABSTRACT. Cardinal invariants related to sequential separability of generalized Cantor cubes , introduced by M. Matveev, are studied here. In particular, it is shownthat the following assertions are relatively consistent with ZFC:
(1)
is sequentially separable, yet there is acountable dense subset ofcontaining no non-trivial convergent subsequence,(2)
is not sequentially separable, yet it is sequentially compact.
$2^{\kappa}$$2^{\omega_{1}}$$2\mathrm{W}1$$2^{\omega_{1}}$The work contained in this paper is devoted to studying combinatorial properties
of independent families and their relationship with sequential separability of generalized Cantor cubes. Connections with -sets and hence the existence of separable
non-metrizable Moore spaces is also mentioned.
Atopological space $X$ is sequentially separable if there is acountable $D\subseteq X$
such that for every $x\in X$ there is asequence $\{x_{n} : n\in\omega\}\subseteq D$ converging to;such a $D\subseteq X$ will be called sequentially dense in $X$ . Aspace $X$ is strongly
sequentially separable if it is separable and every countable dense subset of $X$ issequentially dense. Here we consider sequential separability ofequipped withthe product topology.
Recall that aset
is apseudO-intersection of afamily
if $A\subseteq*F$for every
and,is centered if every non-empty finite subfamily ofhas
an infinite intersection. Afamilyis splitting if$|A\cap S|=|A\backslash S|=\omega$ . A family
is independent provided that for every
nonempty disjoint,.
is independent
splitting if it is both independent and splitting. The following cardinal invariants
are standard.
:
acentered system which has no infinite pseud0$\mathrm{Q}$$x$$2^{\kappa}$$A$$F\in \mathcal{F}$$\mathcal{F}\subseteq[\omega]^{\omega}$$\mathcal{F}$$\mathcal{F}$$\forall A\in[\omega]^{\omega}\exists S\in S$$S\subseteq[\omega]^{\omega}$$\mathrm{I}$$\mathcal{F}_{1}$$\mathfrak{p}$$= \min\{|\mathcal{F}|$intersection}$5= \min$ {:$\mathcal{F}_{2}\in$$\subseteq[\omega]^{\omega}$$[1]^{<\omega}\cap \mathcal{F}_{1}\backslash \cup \mathcal{F}_{2}\neq\emptyset$$\mathrm{I}\subseteq[\omega]^{\omega}$$\mathcal{F}\subseteq[\omega]^{\omega}$

is asplitting family}
F. Tall in [Ta] showed that 2’ is strongly sequentially separable for every.
M. V. Matveev in [Ma] defined the following cardinal invariants (using different
$|S|$

$S$

$\kappa<\mathfrak{p}$notation)$\mathfrak{p}_{1}=\min${$\kappa$:$2^{\kappa}$

is not strongly sequentially separable},

Key words and phrases: sequentially separable space, cardinal invariants of the continuum,
independent families, Q-sets.

The first author’s research was partially supported by agrant$\mathrm{G}\mathrm{A}\check{\mathrm{C}}\mathrm{R}$201/00/1466.
Typeset by $A\mathcal{M}S- IEX$

67
CARDINAL INVARIANTS RELATED TO SEQUENTIAL SEPARABILITY{ :is $= \min$ { :$gg= \min$$2^{\kappa}$$\kappa$$\kappa$

is not sequentially separable} andwith no non-trivial convergent sequences}$\exists D\subseteq 2^{\kappa}$

and observed that
{ : Iis an independent family without apseud0-intersection} and$= \min$ {: Iis an independent splitting family}.
. He asked which of theandHe also noted thatand .
inequalities are consistently strict and what is the relationship between
questions.
these
The main aim of this note is to provide answers to
We assume familiarity with the method of forcing and basic theory of cardinalor [Va]. Set
invariants of the continuum. For reference consult e.g. [BJ], [B1],[Ku].
theoretic notation is standard and follows$\mathfrak{p}_{1}=\min$$i\epsilon$$|\mathrm{I}|$$|\mathrm{I}|$$\mathfrak{p}$$\mathfrak{p}_{1}\leq \mathrm{i}\mathrm{s}$$\leq \mathfrak{p}_{1}\leq\epsilon\epsilon\leq \mathrm{c}$$\leq \mathrm{c}$$z\epsilon$$\mathrm{i}\epsilon$$[\mathrm{v}\mathrm{D}]$I. ZFC results

The proof of the following proposition can be extracted from aconstruction
contained in [DN].

Proposition I.I (Dow-Nyikos).

$\mathfrak{p}_{1}=\mathfrak{p}$.

Proof. Using Matveev’s observation it is sufficient to find an independent family

without apseud0-intersection. To do this fix an independent family I
without apseud0of size and acentered system$=\{(n,m)\in\omega\cross\omega
:
m\leq
n\}$
let
and
intersection. Let A

of size$\{I_{\alpha} :$=$$\mathfrak{p}$\alpha<\mathfrak{p}\}$$\mathcal{F}=\{F_{\alpha} :$\mathfrak{p}$$J_{\alpha}=(F_{\alpha}\cross I_{\alpha})\cap \mathrm{A}$\alpha<\mathfrak{p}\}$.

is an independent family of subsets of
It is easy to see that thatwere apseud0-intersection forwithout infinite pseud0-intersection as if
then $X=proj(Y)$ $=\{n : \exists m(n, m)\in \mathrm{Y}\}$ would be apseud0which is absurd.
intersection of$\{J_{\alpha} :\alpha<\mathfrak{p}\}$$\mathrm{Y}\subseteq\Delta$$\triangle$$\{J_{\alpha} :\alpha<\mathfrak{p}\}$$\{F_{\alpha} :$\square$\alpha<\mathfrak{p}\}$

We will use the following characterization of . We attribute it to folklore as
we do not know how to credit it appropriately. It definitely owes to work of Tall,.
let $A^{0}=A$ andPrzymuszynski, Fleissner and others. For$\epsilon\epsilon$$A^{1}=\omega$$A\subseteq\omega$$\backslash A$

Proposition 1.2 (Folklore). The following are equivalent:(1):} has an(2) There is an independent family Iof size such that {I,infinite pseudO-intersection for every(3) There is a -set of size ,(4) There is a separable norrmal Moore space with closed discrete subset of size$\kappa<\epsilon\epsilon$$\kappa$$I\in \mathrm{I}$$f\in 2^{\mathrm{I}}$$Q$,(5) There is a nomal$\kappa$$\kappa$$\Psi$-space of size .
$\kappa$

For proof of the non-trivial implications see e.g. [GKL]. We will need the following easy fact in the next section

68
MICHAEL

Proposition 1.3. If$\kappa<\epsilon\epsilon$$\mathrm{H}\mathrm{R}\mathrm{U}\check{\mathrm{S}}\acute{\mathrm{A}}\mathrm{K}$then

AND JURIS$2^{\kappa}=2^{\omega}$$\mathrm{S}\mathrm{T}\mathrm{E}\mathrm{P}\mathrm{R}\overline{\mathrm{A}}$NS.

there is countable $D$ sequentially dense subset of . Now, forevery
fix asubset $Df$ of $D$ which converges to . This obviously defines a
one-t0-0ne map frominto.

Proof. As$\kappa<\epsilon\epsilon$$2^{\kappa}$$f\in 2^{\kappa}$$f$$2^{\kappa}$

Corollary 1.4.

$\square$$\mathcal{P}(D)$implies$2^{\omega}<2^{\omega_{1}}$$\epsilon\epsilon=\omega_{1}$.

It is not quite obvious that the cardinal invariant is well defined. In fact, thequestion of existence of an independent splitting family appeared on A. Miller’s
problem list, where he attributes it to K. Kunen. As it turns out the question was
answered along time ago by P. Simon. We include asimple construction of an
independent splitting family here.
$\mathrm{i}\epsilon$

Proposition 1.5. (P. Simon). There is an independent splitting family in $ZFC$.

Proof. Let

be an enumeration of abasis for the topology of therationals Q. Let $\{D_{n} : n\in\omega\}$ be adisjoint refinement of the family,
and let, for eachbe an independent family of subsets of.
Let for$\{U_{n} :n\in\omega\}$$\{U_{n} :$n\in\omega$$\{I_{\alpha}^{n} :\alpha<\mathrm{c}\}$n\in\omega\}$$D_{n}$$\alpha<\mathrm{c}$$J_{\alpha}’=\cup I_{\alpha}^{n}n\in\omega$.

It is easy to see that the family
is independent family of subsets ofand, moreover, all combinations of its elements are dense subsets of Q.
Enumerate all infinite nowhere dense subsets of as
and split
eachinto two infinite subsets. Finally let
and$\{J_{\alpha}’ :\alpha<c\}$$\mathbb{Q}$$\{K_{\alpha} :$\mathbb{Q}$$K_{\alpha}$$M_{\alpha}$\alpha<c\}$$N_{\alpha}$$J_{\alpha}=(J_{\alpha}’\backslash M_{\alpha})\cup N_{\alpha}$

It is obvious thatis still independent as we have made only nowhere
dense changes to dense sets and it is splitting as every infinite subset of contains
for some.
$\{J_{\alpha} :\alpha<c\}$$\mathbb{Q}$$K_{\alpha}$$\alpha<\mathrm{c}$$\square$

While 5 is anatural lower bound on the minimal size of an independent splitting family, as of now, there seems to be adefinite lack of upper bounds. In an
unpublished note [Ny], P. Nyikos proved that assuming the existence of ascale, the dominating number 0is an upper bound. Anatural question arises
as to whether the assumption $=V$ is necessary. However, we do not even know,whether
is relatively consistent with ZFC.
Next we will point out some of the obstacles one would face trying to prove theabove consistency result. According to [KW] call afamily
splitting if
for every sequence $\{A: : i\in\omega\}$ of infinite subsets of there is a $S\in S$ such that$|S\cap A:|=|A:\backslash S|=\aleph_{0}$ for every, and denote by
the minimal cardinalityof ansplitting familyl. Note thatis the uniformity (non(J)) for theideal $J$ on
generated by the sets $I_{A}=$ {or}, whereis an infinite -infinite subset of . Recall that asetis ageneralized$J$ -Luzin set if $X$ is uncountable and for every $A\in J$ $|X\cap A|<|X|$ .
$(\mathrm{b} =\not\supset)$$\mathrm{b}$$5<i\epsilon$$S\subseteq[\omega]^{\omega}\aleph_{0}$$\omega$$i\in\omega$$\mathrm{c}\mathrm{o}$$\sigma-$$B\subseteq\omega:B\subset A$$P(\omega)$$A$$\aleph_{0}-\epsilon$$\aleph_{0}-\epsilon$$\aleph_{0}$$\omega$$B\cap A=\emptyset$$X\subseteq P(\omega)$lThe consistency of $5<\aleph 0-\epsilon$ is another open problem (see [KW] and [Br2]69
CARDINAL INVARIANTS RELATED TO SEQUENTIAL SEPARABILITY

Proposition 1.5. If there is a generalized $J$ -Luzin set of sizethen$\mathfrak{p}$$\mathrm{i}\epsilon$$=\epsilon$.

ofof size one can findthenthe same cardinality which forms an independent family. It should be obviousthat is indeed an independent splitting family (a subset of ageneralized J-Luzin
set of the same cardinality is itself ageneralized $J$-Luzin set). Fix ageneralized
by induction as follows:$J$ -Luzin set $X$ of sizeand construct afamily
$X$
be
let
be any infinite -infinite element of . At stage
Let
be an infinite pseud0-intersection ofa(sequentially) dense subset of 2’ and let. Now,. Such an exists, as
for every
the family.
for everysuch thatpick
It is easy to verify that the family thus constructed is indeed independent.

Proof. We will show that given a -Luzin set
$J$

$X$

$\mathrm{Y}\subseteq X$$\mathfrak{p}$$\mathrm{Y}$$\{I_{\alpha} :\alpha<\mathfrak{p}\}$$\mathfrak{p}$$I_{0}$$\{f_{\dot{1}} :$\alpha<\mathfrak{p}$$\mathrm{c}\mathrm{o}$i\in\omega\}$$A_{:}$$\{I_{\beta}^{f\dot{\cdot}(\beta)} :$i\in\omega$\beta<\alpha\}$$A_{:}$$\alpha<\mathfrak{p}$$i\in\omega$$|I_{\alpha}\cap A_{:}|=|A:\backslash I_{\alpha}|=\aleph_{0}$$I_{\alpha}\in X$$=\mathfrak{p}_{1}$$\square$Corollary 1.6. If there is a Luzin or Sierpiriski set then$i\epsilon$$=\omega_{1}$.

Proof. It suffices to note that the ideal J defined above is -generated closed sets$\mathrm{c}\mathrm{r}$of Haar measure zero.

$\square$

II. Consistency results
Here we will include the main results of this paper. Let us first start with someeasy observations.
Theorem II.1.

$Con(\epsilon\epsilon=i\epsilon =\omega_{1}<\mathrm{c})$.

denotes the-generic over , where
standard c.c.c. poset for adding -many Cohen reals. Let. The fact that Iis an
be the generic sequence of Cohen reals added byindependent splitting family follows easily from the fact that aCohen real splits
every infinite subset from the ground model and the fact that the Cohen reals added.
are mutually generic. So.
for an independent family
aname
be
let
To see that$g:\omega_{1}arrow2$
be the generic. Letsuch that
Then there is ancontradiction
towardsfunction added by the next -many Cohen reals. Assume:
that there is an $A\in V[G]$ which is an infinite pseud0-intersection of {such that. There is

Proof. Let $V\models CH$ and let

$G$

be

$V$

$\mathbb{C}_{\omega_{2}}$$\mathbb{C}_{\omega_{2}}$$\mathrm{I}$$\aleph_{2}$$=\langle C_{\alpha} :\alpha<\omega_{1}\rangle$$G\cap \mathbb{C}_{\omega_{1}}$$i\epsilon.=\omega_{1}$$\{I_{\alpha} :$\mathrm{X}$$\epsilon\epsilon=\omega_{1}$\alpha<\omega_{1}\}$$\mathrm{I}\in V[G\cap \mathbb{C}_{\alpha}]$$\alpha<\omega_{2}$$\aleph_{1}$$I_{\alpha}^{g(\alpha)}$$\omega_{1}\}$$\alpha<$$\xi\in\omega_{2}$$A\in V[G\cap(\mathbb{C}_{\alpha+\xi}\cross \mathbb{C}_{\omega_{2}\backslash (\alpha+\omega_{1})})]$and hence genericity implies that there is some integer$k$such that$A\not\subset^{*}I_{\alpha+\xi+k}^{g(\alpha+\xi+k)}$.

$\square$

It is clear that random reals would do just as well, so asimilar proof gives a
and add -many(just start with amodel ofconsistency ofrandom reals$i\epsilon$$<\mathfrak{y}$$MA\ 2^{\aleph_{\mathrm{O}}}>\aleph_{1}$$\aleph_{2}$70\’AK AND JURISMICHAEL

Theorem II.2. Con($s$<i\epsilon$

Proof. Letand$=\omega_{1}$$\epsilon\epsilon$

5.4).

$V\models$“$5=\omega_{2}$$\mathrm{H}\mathrm{R}\mathrm{U}\check{\mathrm{S}}$$\mathrm{S}\mathrm{T}\mathrm{E}\mathrm{P}\mathrm{R}\overline{\mathrm{A}}$NS).
$2^{\omega_{1}}>\omega_{2}=\mathrm{c}"$. Then$\mathrm{i}\epsilon$$=\omega_{2}$and by Corollary 1.3, Theorem. Such models were constructed along time ago (see e.g.

$[\mathrm{v}\mathrm{D}]$$\square$Consistency offollows directly from results in [D] as pointed out by J.
Brendle in [Br] (see Section III.). We will produce another model for this inequality
where moreoverand amodel where.
Before doing so, recall the definition of astandard forcing for adding apseud0intersection to agiven filter base . The forcing, denoted byconsists of pairs$(s, F)$ , whereis a finite subset of and, ordered by $(s, F)\leq(t, G)$
if is an initial segment of $s(s \backslash t\cap\max(t)+1=\emptyset)$ and. It is easy
to see thatis c.c.c. (in fact -centered)and that forcing withindeedproduces apseud0-intersection to the filter base .
As shown in Proposition II.1 the standard forcing for adding -many Cohen
reals adds an independent splitting family. Our strategy for proving
is to first add two such families and then diagonalize all “paths” through one ofthem producing awitness toand then show that the other one remains
splitting in the extension, witnessing.
$\mathfrak{p}_{1}<\epsilon\epsilon$$i\epsilon$$\epsilon\epsilon<add(N)$$<\epsilon\epsilon$$=\epsilon$$=\mathrm{i}\epsilon$$\mathcal{F}$$\mathrm{M}(\mathcal{F})$$s$$F\in[\mathcal{F}]^{<\omega}$$\omega$$t$$s\backslash t\subseteq\cap G$$\mathrm{M}(\mathcal{F})$$\sigma$$\mathrm{M}(\mathcal{F})$$\mathcal{F}$$\aleph_{1}$$Con(i\epsilon <\epsilon\epsilon)$$\epsilon\epsilon>\omega_{1}$$\mathrm{i}\epsilon$$=\omega_{1}$Theorem II.2.

$Con(\mathrm{i}\epsilon <\epsilon\epsilon)$

Proof. Let $V\models GCH$ . Let$\mathbb{C}_{\omega_{1}}$-generic over$V$.

$\mathrm{P}$$=\mathbb{C}_{\omega_{1}}*\mathrm{P}_{\omega_{2}}$), where (working in$\mathrm{P}_{\omega_{2}}=\langle \mathrm{P}_{\alpha},\dot{\mathbb{Q}} :$\mathrm{V}[\mathrm{H}]$, where $H$ is\alpha<\omega_{2}\rangle$.

is afinite support iteration such thatand::
. Heredenotes the
Cohen real added byand is a -name for afunction fromto 2. By astandard bookkeeping argument one can ensure that(somewhat loosely speaking) for every-name for afunction from to 2thereis ansuch that.
The forcing satisfies the countable chain condition and has (a dense set of) size
so, in the extension. Let be -generic over $V[H]$ . It follows
immediately from the construction, that the family
is independentand that the family
has an infinite pseud0-intersection for every:, hence by Proposition 1.2:$\mathrm{P}_{0}=\mathbb{C}_{\omega_{1}}$$\omega_{1}\})"$$\dot{A}\rho$$1\vdash_{\mathrm{P}_{\alpha}}" \mathbb{Q}$$\beta \mathrm{t}\mathrm{h}$$\mathrm{P}_{0}$$\alpha=\mathrm{M}(\{\dot{A}_{\beta}^{j(\beta)}$$j$$\beta<$$\mathrm{P}_{\alpha}$$\omega_{1}$$f$$\mathrm{P}_{\omega_{2}}$$|\vdash_{\mathrm{P}_{\alpha}}"\dot{\mathbb{Q}}_{\alpha}=\mathrm{M}(\{\dot{A}_{\beta}^{j(\beta)} :$\alpha<\omega_{2}$$\omega_{1}$\beta<\omega_{1}\})"$$\mathrm{P}$$\omega_{2}$$\mathrm{c}$$=2^{\omega_{1}}=\omega_{2}$$G$$\mathrm{P}_{\omega_{2}}$$\{A_{\beta} :$\{A_{\beta}^{f(\beta)} :$f$\beta<\omega_{1}\}$\beta<\omega_{1}\}$$\omega_{1}arrow 2$

Claim II.3.1.

$\epsilon s$$=\omega_{2}$in $V[H][G]$ .

The rest of the proof is devoted to showing that the independent splitting family
added by $H$ remains splitting in the extension. This will be done by showing that,in $V[H]$ , the Boolean algebra generated byis semi-Cohen i.e. has aclosed
unbounded set of regularly embedded countable subalgebras. Note that this implies
that every real added by is Cohen over $V[H]$ and hence, indeed, preserves thatthe independent family added by $H$ remains splitting. Note that this is actuall$\mathrm{P}_{\omega_{2}}$

$G$

71
CARDINAL INVARIANTS RELATED TO SEQUENTIAL SEPARABILITY-generic over $V[H]$ ) then(quite subtle, as if we denote by $K$ the filter
there are many reals in $V[H][G]$ which are NOT Cohen over $V[H][K]$ .
So in order to finish the proof it is sufficient to prove (working in $V[H]$ ) the
following
Claim II.3.2. Let $M\prec H(\omega_{3})$ be (in $V[H]$ ) an elementary submodel containing“
is a maximal. If
and let”.
is a maximal antichain in, thenantichain in$G\cap \mathrm{P}_{0}$$V[H][K\cap(\mathrm{P}_{0}\cap \mathrm{A}\mathrm{t})]$$\beta<\omega_{2}$$\mathrm{P}_{\omega_{2}}$“$\langle$$q$,$(s, \{\dot{A}_{\xi}^{\dot{f}_{\beta}(\xi)} :$D\subseteq M\cap(\mathrm{P}\beta/\mathrm{P}_{0})$$\models$$\mathrm{P}\beta/\mathrm{P}0$$\mathrm{V}[\mathrm{H}][\mathrm{K}\models" D$$M\cap(\mathrm{P}\beta/\mathrm{P}_{0})"$$p|\vdash$$\mathbb{C}_{\omega_{1}}$\xi\in\Lambda_{0}\cup \mathrm{A}\mathrm{x}\}))\in \mathrm{P}_{\beta+1}$”and$p|\vdash$“$j\beta \mathrm{r}$”$\Lambda_{0}\cup\Lambda_{1}=F$

.
and
and
where
$\max(s)=\max(dom(p(\gamma)))$for
Without loss of generality we can assume thatand extend to a so
. (If necessary, find $n>dom(p(\gamma))$ for
every
that, for $m\leq n$ and$F:\Lambda_{0}\cup\Lambda_{1}arrow 2$$\Lambda_{1}\cap\delta=\emptyset(\Lambda_{1}\cap M=\emptyset)$$\Lambda_{0}\subseteq\delta$$p$$\gamma\in\Lambda_{1}$$\gamma\in\Lambda_{1}$$p-$$\gamma\in\Lambda_{1}$$\overline{p}(\gamma)(m)=\{\begin{array}{l}p(\gamma)(m)\mathrm{i}\mathrm{f}m\in dom(p(\gamma))1\mathrm{i}\mathrm{f}m=n\mathrm{a}\mathrm{n}\mathrm{d}F(\gamma)=00\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$and letDenote by$\overline{s}=s\cup\{n\}.)$$\overline{D}$the downward closure of $D$ and let$D^{*}=${$q\in(\mathrm{P}_{\beta}/\mathrm{P}_{0})\cap M$:$\{\dot{q}’\in\dot{\mathbb{Q}}_{\beta}$:$\langle q,\dot{q}’\rangle\in\overline{D}\}$is dense}.

so, by the inductive hypothesis,
is adense subset of
The setfor some $r\in D^{*}$ andsuch that. Chooseis pre-dense in“. Now, using that $\max(s)=$such that, extend to a so that$\max(dom(p(\gamma)))$ for every$(\mathrm{P}_{\beta}/\mathrm{P}_{0})\cap M$$D^{*}$$D^{*}$$(b, \Gamma)\in\dot{\mathbb{Q}}_{\beta}$$\overline{q}\leq r$$\overline{q}\leq q$$\mathrm{P}_{\beta}/\mathrm{P}_{0}$$\langle\overline{q}, (b, \Gamma)\rangle\leq\langle q, (s, \Lambda_{0})\rangle"$$p|\vdash$$\gamma$$\in\Lambda_{1}$$\overline{p}|\vdash$

$p$

$p-$

”

“
$b\backslash s\subseteq\cap\gamma\in\Lambda_{1}\dot{A}_{\gamma}^{F(\gamma)}$by letting$\overline{p}(\gamma)(m)=\{\begin{array}{l}p(\gamma)(m)\mathrm{i}\mathrm{f}m\in dom(p(\gamma))1\mathrm{i}\mathrm{f}m\in b\backslash s\mathrm{a}\mathrm{n}\mathrm{d}F(\gamma)=00\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$for$\in\Lambda_{0}\cup\Lambda_{1}$$\gamma$and $m \leq\max(b)$ . Then$\overline{p}\mathrm{I}\vdash$“ $\langle\overline{q}, (b, \Gamma)\rangle\leq\langle q, (s, \Lambda_{0}\cup\Lambda_{1})\rangle$”

72
MICHAEL$\mathrm{H}\mathrm{r}\mathrm{t}\mathrm{U}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{K}$AND JURIS STEPRANS

which is what we needed to prove.

limit. Fix $D$ and let. Then there is a andsuch.
$p’|\vdash"q$that. The conclusion then follows easily as the set $D^{*}=\{q\in$:is dense} is dense in,hence by the induction hypothesis pre-dense in.
$\beta$$p’$$p\mathrm{I}\vdash"\dot{q}\in \mathrm{P}\beta/\mathrm{P}_{0}"$$\gamma<\beta$$\in \mathrm{P}_{\gamma}’$$(\mathrm{P}_{\gamma}/\mathrm{P}_{0})\cap M$$\{\dot{q}’\in M\cap(P\beta/\mathrm{P}_{\gamma}) :\langle q,\dot{q}’\rangle\in\overline{D}\}$$M\cap(\mathrm{P}7/\mathrm{P}\mathrm{o})$$\mathrm{P}_{\gamma}/\mathrm{P}_{0}$$\square$

The same conclusion probably also holds in amodel constructed in [FM]. Notethat the model constructed in Theorem II.3 shares many of the properties of theCohen model (model used in Theorem II.1). For instance,, andnon(M)
in fact there is aLuzin set, in both models.
Next we will show, that not onlyis consistent (as shown by Theorem II.2)
but it is possible to lower while leaving “most” cardinal invariants large. Recall
that antree
is aSuslin tree if it has no uncountable chains or antichains.
When used as aforcing notion(turned up-side-down) is ac.c.c partial order“$MA_{\sigma-centered}+$which is does not add any new reals. We will show that if$V$
There is aSuslin tree ”and if is -generic over then $V[G]\models gg$. Notethat such amodel can be obtained e.g. by adding asingle Cohen real to amodel of
MA (see [BJ]). In this model, however, add(N) $=cov(N)$(see [BJ]). Thereis away, however, to construct amodel ofis aSuslin tree
where, moreover, add.
$cov(\mathcal{M})=\mathrm{c}$$=\omega_{1}$$\mathit{5}S$$<\mathrm{i}\epsilon$$\epsilon\epsilon$$T$$\omega_{1}$$T$$V\models$

$T$

$G$

$T$

$=\omega_{1}$$=\omega_{1}$$MA_{\sigma-centered}+\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$(Af)=\mathrm{c}$Theorem II.4. Cm $(\epsilon\epsilon<add(N)=5=\mathrm{c})$ .

Proof. Let$V\models MA_{\sigma-centered}$, add(N)$=\mathrm{c}$$+\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$is aSuslin tree

$T$

. Let

$G$

be

-generic over .
First note that as
is c.c.c. all cardinalities and cofinalities of ordinals arepreserved. Moreover, as $.|T|=\omega_{1}$ ,. To see that
note that,since adds no reals, if were aname for asplitting family of size less than thenthe family
$T$

$V$

$T$

$V[G]\models \mathrm{c}$$T$$=\mathrm{c}^{V}$$5=\mathrm{c}$

$S$

$c$

$\{S\in[\omega]^{\omega} :\exists t\in T t|\vdash" S\in\dot{S}"\}$

would be asplitting family of size less than in , which contradicts $MA_{\sigma-centered}$ .
Asimilar proof gives $V[G]\models add(N)=\mathrm{c}$ .
So the only thing left to prove is that, in other words, no countabledense subset ofsequentiallyisdense. Without loss of generality we can assume
thatand that it is well-pruned and everywhere branching (i.e.
,. While adds no new reals it does add new elements of . Aiming
towards acontradiction let 4be a -name for afunction ffom toand assume“
thatis sequentially dense. Let 0be aregular cardinal such that$T\in H(\theta)$ . Put$\mathrm{c}$$\epsilon\epsilon$$V$$=\omega_{1}$$2^{\omega_{1}}$$T\subseteq 2^{<\omega_{1}}$$t^{\wedge}\mathrm{O}$$t^{\wedge}1\in T)$$\forall t\in T$$T$$2^{\omega_{1}}$$T$$|\vdash_{T}$$\omega$$2^{\omega_{1}}$$rng(\dot{\Phi})$$C$$=\{M\cap\omega_{1} :M\prec H(\theta)\ \dot{\Phi}, T\in M\}$and increasingly enumerate as. Now let”element ofdefined by
if and only if
To finish the proof it is sufficient to show that$C$

$2^{\omega_{1}}$$\{\delta_{\alpha} :$p|\vdash"\dot{f}(\alpha)=i$\alpha<\omega_{1}\}$

$j$

be a

$\delta_{\alpha}\in dom(p)$$T$

name for anand$\mathrm{p}(\mathrm{S}\mathrm{a})=i$.

73
CARDINAL INVARIANTS RELATED TO SEQUENTIAL SEPARABILITY

Claim.

$|\vdash_{T}$“There is no sequence in$rng(\dot{\Phi})$conveyying to j”.

.

such that
Assume the contrary, i.e. there is a $p\in T$ and an$j”$ . Note that this is without loss of generality as
does not add any new reals.
$\Phi\inM$$M\prec
H(\theta)$,,
and let $\rho=M\cap\omega_{1}$ . Thensuch that
Now, however, pick
,
.
. Let
loss of generality
and without
for some.
, yet it does not decide thefor every
Then decides the value of“. Now, it is easy to reach acontradiction; If thereasvalue ofsuch that $q|\vdash"\dot{\Phi}(n)(\alpha)=0$”let $q’=q^{\wedge}1$ otherwise let
are infinitely many”
which is absurd.
. In either case$p|\vdash"\Phi$ $\circ harrow$$h\in\omega^{\omega}$

$T$

$p$

$T$

$\alpha<\rho$$\alpha\leq\rho$$\rho=\delta_{\alpha}$$\Phi$$q$$j(\alpha)$$q^{\wedge}i\mathrm{I}\vdash$$(n)(\alpha)$$q\in T_{\rho}$$q\supset p$$n\in\omega$$j(\alpha)=\mathrm{i}\mathrm{n}$$n’ \mathrm{s}$$q’|\vdash"\dot{\Phi}\circ h\star j$$q’=q^{\wedge}\mathrm{O}$$\square$Let us point out the curious nature of the cardinal invariant . It is proved
yet all invariants(though not stated) in [JS] that it is consistent that
II.4 shows that
hand,
Theoremfrom the Cichon diagram are small. On the otherand most “standard”
consistently $gg<add(N)$ , so there is no relation between
cardinal invariants of the continuum.
$\epsilon\epsilon$$\epsilon\epsilon=\omega_{2}$$\epsilon\epsilon$

REFERENCES
[BJ]

[B1]
[Br]
[Br2]
[vD]

[D]
[DN]

[FM]

T. Bartoszynski and H. Judah, Set Theory, On the Structure of the real line, A
Peters, 1995.
A. Blass, Combinatorial Cardinal Characteristics of the Continuum, to appear in
Handbook of Set theory.
J. Brendle, Dow’s principle and $Q$ -sets, Canad. Math. Bull. 42 (1999), 13-24.
$\mathrm{K}$J. Brendle, Around splitting and reaping, Comment. Math. Univ. Carolin. 39 (1998),
269-279.
E. van Douwen, The integers and topology, in Handbook of Set Theoretic Topology(ed. K. Kunen and J. Vaughan), 1984, pp. 111-167, North-Holland, Amsterdam.
A. Dow, On compact separable radial spaces, Canad. Math. Bull. 40 (1997), 422-432.
A. Dow and P. Nyikos, Representing free Boolean algebra, Fund. Math. 141 (1992),
21-30.
W. G. Fleissner and A. W. Miller, On $Q$ -sets, Proc. Amer. Math. Soc. 78 (1980),

280-284.
[GKL]
[JS]
[KW]
[Ku]
[Ma]
[Ny]
[Ta]
[Va]

P. M. Gartside, R. W. Knight and J. T. H. Lo, Sequential density, (preprint).
H. Judah and S. Shelah, $Q$ sets Sierpitiski sets and rapid filters, Proc. Amer. Math.
Soc. Ill (1991), 821-832.
A. Kamburelis and B. Weglorz, Splittings, Arch. Math. Logic 35 (1996), 263-277.
K. Kunen, Set Theory. An Introduction to Independence Proofs (1980), North Holland, Amsterdam.
M. V. Matveev, Cardinal and a theorem of Pelczynski, (preprint).
$\mathfrak{p}$P. Nyikos, Answer to a question of Matveev, (privat communication).
F. Tall, How separable is a $space^{Q}$ That depends on your set $theory^{/}.$, Proc. Amer.
Math. Soc. 46 (1974), 310-314.
J. Vaughan, Small uncountable cardinals and topology, In J. van Mill and G. M. Reed,editors, Open problems in topology (1990), North-Holland, 196-216.

Faculteit der Exacte Wetenschappe$\mathrm{n}$74
MICHAEL$\mathrm{H}\mathrm{R}\mathrm{U}\check{\mathrm{S}}$\’AK AND JURIS

Vrije Universiteit
De Boelelaan
1081 HV Amsterdam, The Netherlands$1081\mathrm{a}$michael@cs.vu.nl
Dept. of Math, and Stat.
York University

4700 Keele Street
1P3 Toronto, Canada$\mathrm{M}3\mathrm{J}$steprans@pascal.math, yorku. $ca$$\mathrm{S}\mathrm{T}\mathrm{E}\mathrm{P}\mathrm{R}\overline{\mathrm{A}}$NS