UDC 519.50

MATHEMATICS

I. D. STUPINA

ON SOME PROPERTIES OF (R)-, (R^c)-OPERATIONS AND PROJECTIVE OPERATIONS ON UNCOUNTABLE SYSTEMS OF SETS IN CONNECTION WITH BRANCHING HYPOTHESES

(Presented by Academician M. A. Lavrent'ev, 12 I 1970)

In the paper ((^1)) we considered certain properties of (R)- and (R^c)-operations on countable systems of sets. Analogous properties of certain operations were considered in ((^2)). In the present note analogous questions are considered for the same operations on uncountable systems of sets. We use the notation introduced in note ((^1)).

1. Let (T=\langle \mathcal E,<\rangle) be a branching table, (U\subset \mathcal E), (T^=\langle U(T),<\rangle). If (\rho(T^)=\omega_\lambda), (\omega_\tau) is the final character of the number (\omega_\lambda), and (F=(x_\beta){\beta<\omega\lambda}) is a monotone dissection of the table (T'), then ((\forall \beta<\omega_\lambda)\cdot [\overline{(x_\beta,\cdot){T^}\cap U}\ge \aleph_\tau]), and further either (\overline{F}\cap \overline{U}=\aleph_\tau), or there exists a disjunctive subset (K\subset U) such that ((\forall \beta<\omega_\lambda)[\overline{(x_\beta,\cdot)_{T^}\cap K}\ge \aleph\tau]).

In what follows, by (\aleph_\nu) we shall denote a strongly inaccessible cardinal number. If (\overline E=p), then (E) will be called a (p)-set. The following hypotheses are used in the paper:

((\alpha_1)) Every branching table (T) of rank (\omega_\nu) such that ((\forall \alpha<\omega_\nu)\cdot[\overline{T_\alpha}<\aleph_\nu]) attains its rank.

((\alpha_2)) If in a branching table (T=\langle \mathcal E,<\rangle) of rank (\omega_\nu) the cardinality of every disjunctive subset (U\subset \mathcal E) is less than (\aleph_\nu), then the table (T) attains its rank ((^3)).

These hypotheses are respectively equivalent to the following:

((\beta_1)) Every branching table (T) of rank (\omega_\nu) either attains its rank or has an (\aleph_\nu)-node.

((\beta_2)) Every branching table (T=\langle \mathcal E,<\rangle) of rank (\omega_\nu) either attains its rank or satisfies the condition: there exists a disjunctive (\aleph_\nu)-subset (U\subset \mathcal E) not satisfying condition (i_\nu).

Obviously, ((\alpha_1)\Rightarrow(\alpha_2)). Theorems proved with the aid of the hypotheses ((\alpha_1)) and ((\alpha_2)) will be marked by (()) and ((*)), respectively.

((*)) For every (\aleph_\nu)-set (U\subset \mathcal E) in the table (T=\langle \mathcal E,<\rangle) of rank (\le \omega_\nu), (I_\nu) or (II_\nu) is valid.

1. By the (\Pi^\tau)-product of discrete spaces ({X_\alpha:\alpha<\omega_\tau}), denoted by us as (\Pi^\tau X_\alpha) and called a generalized Baire space, we shall mean the space on the abstract product

[
\prod_{\alpha<\omega_\tau} X_\alpha,
]

whose open-closed base (B^\tau) is given by the totality of generalized Baire intervals (\delta_{(i_\beta)_{\beta<\alpha}}), where

[
\delta_{(i_\beta){\beta<\alpha}}
=
\left{(j\beta){\beta<\omega\tau}\in
\prod_{\beta<\omega_\tau}X_\beta:
(\forall \beta<\alpha)\,[j_\beta=i_\beta]\right}.
]

We shall assume

[
\delta_{(i_\beta){\beta<0}}
=
\prod X_\alpha.
]

If ((\forall \alpha<\omega_\tau)\,[X_\alpha=J]) and (J\ne \aleph_\tau), then the space (\Pi^\tau X_\alpha) is denoted by (J^{\omega_\tau}). Put (T^\tau=\langle B^\tau,<\rangle), where as the relation (<) the relation (\supset) of strict inclusion is taken.

((*)) In order that a closed set (E\subseteq \Pi^\tau X_\alpha) not be (\aleph_\nu)-bicompact, it is necessary and sufficient that it have a disjunctive

an $\aleph_\nu$-covering $S \subset B^\nu$, satisfying condition II$\nu$, i.e., such that the branching table $\langle S(T^\nu), <\rangle$ has an $\aleph\nu$-node.

Remark. I. I. Parovichenko ($^4$) proved that hypothesis $(a_1)$ is equivalent to the assertion of $\aleph_\nu$-bicompactness of the $T^\nu$-product of discrete spaces ${X_\alpha:\alpha<\omega_\nu}$, where $(\forall \alpha<\omega_\nu)[\overline{\overline{X}}\alpha<\aleph\nu]$. Since the classes of open sets in this space and in the space $\prod^\nu X_\alpha$ coincide, hypothesis $(a_1)$ is equivalent to the assertion of $\aleph_\nu$-bicompactness of the space $\prod^\nu X_\alpha$, when $(\forall \alpha<\omega_\nu)[\overline{\overline{X}}\alpha<\aleph\nu]$.

1. The set of all tuples of the form $(i_\beta){\beta<\alpha}$, corresponding to all points $x\in J^\nu$, will be denoted by $W$. A set $U\subset W$ will be called a $W$-base. Unless special qualifications are made, the $R$- and $R^c$-operations are considered with the full depth of chains $\upsilon=\omega\nu$. The rigid $W$-bases of the operations $R_{\mathfrak M}$, $R_{\mathfrak M}^c$, $R_N^\alpha$, $R_N^{\alpha c}$, $R^\alpha$, $R^{\alpha c}$, $0\leqslant\alpha<\omega_{\nu+1}$ ($^5$, $^6$) will be denoted respectively by $\theta_{\mathfrak M}$, $\theta_{\mathfrak M}^c$, $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $\theta^\alpha$, $\theta^{\alpha c}$. By $\chi_\alpha$ [$\chi_\alpha^c$] we denote the rigid base of the projective $A_\alpha$ [$CA_\alpha$]-operation, $0\leqslant\alpha<\omega_{\nu+1}$.

2. We shall say that a rigid base $N$ admits a $V$-transformation if, for every $J'\subset J$, under the condition $N^{J'}\ne\varnothing$, there exists a set $[J']$ such that, putting for an arbitrary system of sets $(E_i)$ $E_i=E_i'$, if $i\notin [J']$, and $E_i=\varnothing$, if $i\in [J']$, we obtain $\Phi_N^{J'}(E_i)=\Phi_N(E_i)$.

We have proved: if $N$ and $N^c$ are rigid bases of mutually complementary operations, then each of them admits a $V$-transformation.* Therefore the bases $\chi_\alpha$, $\chi_\alpha^c$, $0\leqslant\alpha<\omega_{\nu+1}$, and also the bases $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $0\leqslant\alpha<\omega_{\nu+1}$, under the condition that $N$ and $N^c$ are rigid bases, admit a $V$-transformation.

A rigid base $N$ is called $\aleph_\tau$-regular ($^8$) for a class of sets $\mathcal K$ if $(\forall J'\subset J)[\Phi_N^{J'}(\mathcal K)\subset\Phi_N(\mathcal K)]$ and the class of sets $\Phi_N(\mathcal K)$ is invariant with respect to the operations $\underset{\aleph_\tau}{\cup}$, $\underset{\aleph_\tau}{\cap}$. A property $H$ of chains of a rigid base $N$ is called $N$-regular ($^8$) for a class of sets $\mathcal K$ if the operations $\Phi_{HN}$, $\Phi_{(HN)^i}$ are weaker than the operation $\Phi_N$ with respect to the class of sets $\mathcal K$.

Each of the bases $\chi_\alpha$, $\chi_\alpha^c$, $0\leqslant\alpha<\omega_{\nu+1}$, and also each of the bases $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $1\leqslant\alpha<\omega_{\nu+1}$, when the conditions are fulfilled: $1^\circ$. $N$ and $N^c$ are rigid bases; $2^\circ$. The operation $\Phi_N$ is stronger than the operation $\underset{\aleph_\nu}{\cup}$, is $\aleph_\nu$-regular for the class of sets $\mathcal K\ni\varnothing,\Xi$, where $\Xi$ is the basic space.

Hence, and by virtue of Theorem 3 of I. Kozlova (($^8$), Theorem 1), for the class of sets $\mathcal K\ni\varnothing,\Xi$, for every $p\leqslant\aleph_0$ the property $H_x$ is $M$-regular, if $M$ is a base $\chi_\alpha$, $\chi_\alpha^c$, $0\leqslant\alpha<\omega_{\nu+1}$, and also a base $\theta_N^\alpha$, $\theta_N^{\alpha c}$, $1\leqslant\alpha<\omega_{\nu+1}$ in the case when conditions $1^\circ$ and $2^\circ$ are fulfilled, and, consequently, when $M$ is a base $\theta^\alpha$, $\theta^{\alpha c}$, $1\leqslant\alpha<\omega_{\nu+1}$.

1. For a base $N$ define the $\Delta\Sigma$-operation $Q_N$ over two systems of sets $(E_u)$, $(e_v)$, by putting

[
Q_N(E_u,e_v)=
\bigcup_{u,v}\ \bigcup_{\xi\in N,\ \xi'\subset\xi,\ \overline{\overline{\xi'}}=\aleph_\nu}
\left(\bigcap_{u\in\xi} E_u\cap \bigcap_{v\in\xi'} e_v\right).
]

(*) For any $0\leqslant\alpha<\omega_{\nu+1}$ the operation $Q_{\theta^\alpha}$ [$Q_{\theta^{\alpha c}}$] is weaker than the operation $\Phi_{\theta^\alpha}$ [$\Phi_{\theta^{\alpha c}}$] with respect to the class of sets $\mathcal K\ni\varnothing,\Xi$.

1. Let $\mathfrak M=(N_a){a\in W}$, $\mathfrak M^c=(N_a^c)$, we introduce thinning conditions. We shall say that a collection of tuples $\eta_x$ satisfies condition $a_1^\nu$ [$a_3^\nu$] if in the branching table $\langle\eta_x(T_W^\nu),<\rangle$ there is a node $((a_i))}$ be tables of rigid bases. For the operations $\Phi_{\theta\mathfrak M}$, $\Phi_{\theta\mathfrak M^c{i\in J'}$ from which one can form a $>\aleph\nu$ [$\geqslant\aleph_\nu$] $R$-($R^c$-) covering of the tuple $a$; $a_2^\nu$, if in the collection $\eta_x(T_W^\nu)$ there is a tuple and such an $R$-($R^c$-) covering of it in which $\aleph_\nu$ tuples have the property of bisection; $a_4^\nu$, if the table $\langle\mu_x,<\rangle$ attains its rank $\omega_\nu$; $a_7^\nu$, if the collection $\mu_x$ includes a disjunctive $\aleph_\nu$-subset of tuples.

* A. D. Taimanov ($^7$) gave an example of a $\delta S$-operation with a rigid base for which the complementary $\delta S$-operation has no rigid base.

Then for the operations (\Phi_{\theta_{\mathfrak M}}, \Phi_{\theta_{\mathfrak M}^{c}}) the following holds:

1) for (\nu<\omega_\nu):
[
\overline{M}x>\mathfrak N\nu \Longleftrightarrow a_1^\nu \vee a_2^\nu,\qquad
\overline{M}x\geq \mathfrak N\nu \Longleftrightarrow a_2^\nu \vee a_3^\nu,\qquad
\overline{M}x=\mathfrak N\nu \Longleftrightarrow \neg(a_1^\nu\vee a_2^\nu)\ \&\ a_3^\nu;
]

2) ((**))
[
\overline{M}x\geq \mathfrak N\nu\ \&\ \neg a_3^\nu\ \&\ \neg a_7^\nu \Rightarrow a_4^\nu;
]

3) ((**)).
[
\overline{\overline{M}}x\geq \mathfrak N\nu \Longleftrightarrow a_3^\nu\vee a_4^\nu\vee a_7^\nu.
]

For the operation (\Phi_{\theta_{\mathfrak M}^{c}}) the following holds:

4) ((*))
[
\overline{\overline{M}}x>\mathfrak N\nu \Longleftrightarrow a_1^\nu\vee a_2^\nu;
]

5) ((*))
[
\overline{\overline{M}}x=\mathfrak N\nu \Longleftrightarrow \neg(a_1^\nu\vee a_2^\nu)\ \&\ a_3^\nu.
]

1. Let the (W)-base (U) coincide with the base (\theta_{\mathfrak M}) or (\theta_{\mathfrak M}^{c}). For (k=2,4,7) put
   [
   \Phi_{H_{a_k}U}(E_a)={x\in\Phi_U(E_a):\ \text{the set }\eta_x\text{ satisfies the condition }a_k^\nu}.
   ]

((*)) For the class of sets (\mathcal K\ni\varnothing,\ \Xi), the operations
[
\Phi_{H_{a_2}\theta^\alpha},\quad
\Phi_{H_{a_2}\theta^{\alpha c}},\quad 0\leq \alpha<\omega_{\nu+1},
]
are weaker than the operations (\Phi_{\theta^\alpha}, \Phi_{\theta^{\alpha c}}), respectively.

((**)) If conditions (1^0) and (2^0) are fulfilled, the operations
[
\Phi_{H_{a_4}\theta_N^\alpha},\quad
\Phi_{H_{a_7}\theta_N^\alpha},\quad 2\leq \alpha<\omega_{\nu+1},
]
are weaker than the operation (\Phi_{\theta_N^\alpha}), and the operations
[
\Phi_{H_{a_4}\theta_N^{\alpha c}},\quad
\Phi_{H_{a_7}\theta_N^{\alpha c}},\quad 1\leq \alpha<\omega_{\nu+1},
]
are weaker than the operation (\Phi_{\theta_N^{\alpha c}}).

((**)) If conditions (1^0) and (2^0) are fulfilled, the property (H_{\mathfrak N_\nu}) is (M)-regular if the base (M) coincides with the base (\theta_N^\alpha,\ 2\leq\alpha<\omega_{\nu+1},\ \theta_N^{\alpha c},\ 1\leq\alpha<\omega_{\nu+1}), in particular, with the base (\theta^\alpha) for (2\leq\alpha<\omega_{\nu+1}), and (\theta^{\alpha c}) for (1\leq\alpha<\omega_{\nu+1}).

((*)) The property (H_{\widehat{\mathfrak Z}_\nu}) is (\theta^{\alpha c})-regular for the class of sets (\mathcal K\ni\varnothing,\Xi).

1. Let (J={\alpha:\alpha<\omega_\nu}). If each chain (\xi) of the rigid base (M) is ordered in the order of increase of its elements, then we obtain a rigid reduced base (\breve M). Let (\breve M\subset J^{\omega_\nu}). Put
   [
   \Phi_{H_c\breve M}(E_i)
   =
   {x\in\Phi_M(E_i):\ \text{the closure of the set }\breve M_x\text{ is not }\mathfrak N_\nu\text{-bicompact}},
   ]
   where (\breve M_x) is the set of all chains of the base (\breve M) whose kernels contain the point (x). Let
   [
   q=(i_\beta){\beta<\alpha}\in W
   \quad\text{and}\quad
   (\beta'<\beta'')\Rightarrow (i