UDC 519.52

Z. I. KOZLOVA

AN \(R\)-OPERATION WITH FULL DEPTH OF CHAINS OVER SYSTEMS OF SETS OF CARDINALITY \(\tau\)

(Presented by Academician M. A. Lavrent’ev on 12 I 1970)

Let \(V\) be the totality of all ordinal numbers; \(K_{\mathrm I}\) the totality of indefinite ordinal numbers, \(K_{\mathrm{II}}\) the limiting ordinal numbers; \(I=\{i\mid i\in V\ \&\ i<\omega_\nu\}\), \(\tau=\aleph_\nu\) a strongly inaccessible cardinal number; \(Y_\gamma=\langle i_0,\ldots,i_a,\ldots\mid\gamma\rangle\) a tuple of rank \(\gamma\), where \(\gamma<\omega_\nu\) and \((\forall a<\gamma)[i_a\in I]\); \(W=\{Y_\gamma\mid\gamma<\omega_\nu\}\) the space of tuples \((\operatorname{Card} W=\tau)\); \(\eta\subseteq I\), \(\vartheta\subseteq W\) chains; \(\Xi=\{\eta\mid\eta\subseteq I\}\), \(\Xi^*=\{\vartheta\mid\vartheta\subseteq W\}\) spaces of chains; \(N\subseteq\Xi\), \(M\subseteq\Xi^*\) bases; \(\mathfrak R\) the basic space, whose subsets are studied.

Put \(Y_\gamma,\ i_\gamma=\langle i_0,\ldots,i_a,\ldots,i_\gamma\rangle\), if \(Y_\gamma=\langle i_0,\ldots,i_a,\ldots\mid\gamma\rangle\);
\[ Y_\gamma Y_{\gamma'}=\langle i_0,\ldots,i_a,\ldots,i_{\gamma'},\ldots,i^\alpha,\ldots\mid\gamma+\gamma'\rangle, \]
if \(Y_\gamma=\langle i_0,\ldots,i_a,\ldots\mid\gamma\rangle\),
\[ Y_{\gamma'}=\langle i_\gamma,\ldots,i_{\gamma+\alpha},\ldots\mid\gamma'\rangle. \]
We shall say that \(Y\) is subordinate to \(Y'\), and write \(Y'<Y\), if
\[ Y=\langle i_0,\ldots,i_a,\ldots\mid\gamma\rangle,\qquad Y'=\langle i_0,\ldots,i_a,\ldots\mid\gamma'\rangle \]
and \(\gamma'<\gamma\); \(Y'\leq Y\), if \(Y'<Y\) or \(Y'=Y\). The rank of \(Y\) is denoted by \(\rho(Y)\).

A. A. Lyapunov \((^1)\) considered an \(R\)-process of transfinite transformation of sets of a given family of arbitrary cardinality, starting from a given system of \(\Delta\Sigma\)-operations. The depth of the chains of this \(R\)-process is equal to \(\omega\). We shall study an \(R\)-process of transfinite transformation of sets of a given family of cardinality \(\tau\), starting from a given system of \(\Delta\Sigma\)-operations, when the depth of the chains of the \(R\)-process is full, i.e. equal to \(\omega_\nu\).

Let
\[ \mathfrak R=(N_Y)_{Y\in W},\qquad N_Y\subseteq\Xi, \]
\((E_Y)_{Y\in W}\) be an arbitrary family of sets of the space \(\mathfrak R\), in which
\[ E_{Y_\gamma}=\bigcap_{Y'<Y_\gamma}E_{Y'} \quad\text{for }\gamma\in K_{\mathrm{II}}. \]

By the \(\nu\)-conjunctive extension of \(\Delta\Sigma\)-operations with bases of the family \(\mathfrak R\), where \(\nu=\omega^\lambda\leq\omega_\nu\), we call the family of \(\Delta\Sigma\)-operations whose bases
\[ (M_Y^\lambda)_{Y\in W}=\mathfrak M(\nu) \]
are defined as follows:
\[ \mu\in M_Y^\lambda\equiv(\rho(Y)\in K_{\mathrm I}\&Y\in\mu\vee\rho(Y)\in K_{\mathrm{II}}\&(\forall Y'<Y)[Y'\in\mu])\& \]
\[ \&(\forall YY'\in\mu)(\exists\eta\in N_{YY'})(\forall i\in\eta)[YY',i\in\mu]\& \]
\[ \&(\forall\gamma<\nu)(\forall Y'<Y_\gamma)[\gamma\in K_{\mathrm{II}}\&YY'\in\mu\to \]
\[ \to(\exists\eta\in N_{YY_\gamma})(\forall i\in\eta)[YY_\gamma,i\in\mu]]. \]

Put
\[ E_Y^0=E_Y,\qquad E_Y^1=R_Y,\qquad \mathfrak M\{E_{Y'}\}=\Phi_{M_Y^1}\{E_{Y'}\}. \]

If the derived sets \((E_Y^\alpha)_{Y\in W}\) are defined, then we construct the family of bases \((N_Y^\alpha)_{Y\in W}\) in the following way: let
\[ Z_Y^\alpha(\eta)=\{Y_{\omega_\alpha}\mid(\forall\beta<\omega^\alpha)[Y_\beta<Y_{\omega_\alpha}\to YY_\beta\in\eta]\}, \]
\[ N_Y^\alpha(\eta)=\{\xi\mid(\exists Y_{\omega_\alpha}\in Z_Y^\alpha(\eta))(\exists\eta_{Y_{\omega_\alpha}}\in N_{YY_{\omega_\alpha}})\times \]
\[ \times[\xi=(YY_{\omega_\alpha},i)_{i\in\eta_{Y_{\omega_\alpha}},\ Y_{\omega_\alpha}\in Z_Y^\alpha(\eta)}]\}; \]
then
\[ N_Y^\alpha=\{\eta\mid(\exists\mu\in M_Y^\alpha)(\exists\xi\in N_Y^\alpha(\mu))[\eta=\mu+\xi]\}. \]
Put
\[ E_Y^{\alpha+1}=R_{Y;\{N_Y^\alpha\}}\{E_Y^\alpha\}. \]

If the derivative sets \((E_Y^\alpha)_{Y\in W}\) are defined for all \(\alpha<\varkappa<\omega_\nu,\ \varkappa\in K_{\mathrm{II}}\), then put \(E_Y^\varkappa=\bigcap_{\alpha<\varkappa}E_Y^\alpha\).

Construct a family of bases \((N_Y^\varkappa)_{Y\in W}\) and put
\[ E_Y^{\varkappa+1}=R_{Y;\{N_Y^\varkappa\}}\{E_Y^\varkappa\}. \]

Thus, we shall construct derivative sets \((E_Y^\alpha)_{Y\in W}\) for all \(\alpha<\omega_\nu\). Put
\[ E_Y^{\omega_\nu}=\bigcap_{\alpha<\omega_\nu}E_Y^\alpha . \]

Repeating the process of differentiating sets, starting from the family of sets \((E_Y^{\omega_\nu})_{Y\in W}\) as the initial family, we obtain derivative sets \((E_Y^{\omega_\nu\cdot 2})_{Y\in W}\). By further repetition of this process we obtain derivative sets \((E_Y^{\omega_\nu\cdot\beta})_{Y\in W}\) for \(\beta<\omega_{\nu+1}\), while taking
\[ E_Y^{\omega_\nu\cdot\varkappa}=\bigcap_{\alpha<\varkappa}E_Y^{\omega_\nu\cdot\alpha}, \]
if \(\varkappa\in K_{\mathrm{II}}\). Let
\[ \vartheta^{\omega_\nu\cdot\alpha}(x)=\{Y\mid x\in E_Y^{\omega_\nu\cdot\alpha}\}. \]
Since the sequence of chains
\[ \vartheta^{\omega_\nu}(x)\supseteq\vartheta^{\omega_\nu\cdot2}(x)\supseteq\cdots\supseteq \vartheta^{\omega_\nu\cdot\alpha}(x)\supseteq\cdots \]
does not increase, \(\operatorname{Card} W=\aleph_\nu\), and \(\operatorname{Card}\omega_{\nu+1}=\aleph_{\nu+1}\), it follows that
\[ (\exists\alpha_0<\omega_{\nu+1})(\forall\alpha>\alpha_0)\times [\vartheta^{\omega_\nu\cdot\alpha_0}(x)=\vartheta^{\omega_\nu\cdot\alpha}(x)] . \]
The stabilized chain \(\vartheta^{\omega_\nu+\alpha_0}(x)\) we shall denote by \(\vartheta^{\omega_{\nu+1}}(x)\).

We shall call the operation
\[ (\omega_\nu)R_{Y;\mathfrak m} \]
over a family of sets \([E_Y)_{Y\in W}\) the \((\omega_\nu)R_{Y;\mathfrak m}\)-operation
\[ (\omega_\nu)R_{Y;\mathfrak m}\{E_Y\} = \bigcap_{\alpha<\omega_{\nu+1}}E_Y^{\omega_\nu\cdot\alpha} = E_Y^{\omega_{\nu+1}} . \]

The family of \(\Delta\Sigma\)-operations
\[ \bigl((\omega_\nu)R_{Y;\mathfrak m}\bigr)_{Y\in W} \]
is the \(\omega_\nu\)-conjunctive extension of \(\Delta\Sigma\)-operations with bases of the family \(\mathfrak M\).

Put
\[ (\nu)R_{Y;\mathfrak m}=\Phi_{M_Y^\lambda}, \]
where \(M_Y^\lambda\in\mathfrak M(\nu)\), \(\nu=\omega^\lambda\leqslant\omega_\lambda\). Let \(\mathfrak C=(P_Y)_{Y\in W}\), where \(P_Y\subseteq\Xi\), and let \(\theta\subseteq\Xi^*,\ Y\in\vartheta\in\theta\). We replace the tuple \(Y\) by the family of tuples
\[ \lambda_Y^1=\{Y;\ i_0\mid (i_0)=\xi\in P_Y\}. \]
If \(Y\) is replaced by \(\lambda_Y^\alpha\), and at the same time at least one of the tuples
\[ YY_\alpha=Y\langle i_0,\ldots,i_{\alpha'},\ldots\mid \alpha\rangle\in\lambda_Y^\alpha \]
is replaced by
\[ \lambda_{YY_\alpha}=\{Y\langle i_0,\ldots,i_\alpha\rangle\mid (i_\alpha)=\xi\in P_{YY_\alpha}\}, \]
then we say that \(Y\) is replaced by \(\lambda_Y^{\alpha+1}\). If \(Y\) is replaced by \(\lambda_Y^\alpha\) for all \(\alpha<\gamma\), then we replace the tuple
\[ YY_\gamma=Y\langle i_0,\ldots,i_\alpha,\ldots\mid \gamma\rangle, \]
where
\[ Y\langle i_0,\ldots,i_{\alpha'},\ldots\mid \alpha\rangle\in\lambda_Y^\alpha \quad\text{for }\alpha<\gamma, \]
by the family of tuples
\[ \lambda_{Y\langle i_0,\ldots,i_\alpha,\ldots\mid\gamma\rangle} = \{Y\langle i_0,\ldots,i_\gamma\rangle\mid (i_\gamma)=\xi\in P_{YY_\gamma}\}, \]
and say that \(Y\) is replaced by the family of tuples \(\lambda_Y^{\gamma+1}\).

By the \(\nu\)-disjunctive extension of \(\Delta\Sigma\)-operations with bases of the family \(\mathfrak C\) we shall mean the family of \(\Delta\Sigma\)-operations whose bases \((\theta_Y)_{Y\in W}=\mathfrak C(\nu)\) are defined as follows:
\[ \vartheta\in\theta_Y\equiv \rho(Y)\in K_{\mathrm{II}}\ \&\ (\exists Y'<Y)[\vartheta=\{Y'\}] \vee \rho(Y)\in K_I\&\vartheta=\{Y\} \vee(\exists\vartheta'\in\theta_Y)(\exists Y'\in\vartheta')(\exists\gamma<\nu)(\exists\lambda_{Y'}^{\gamma+1}) [\vartheta=(\vartheta'\setminus\{Y'\})\cup\lambda_{Y'}^{\gamma+1}] . \]

The operations supplementary to the \((\omega_\nu)R_{Y;\mathfrak m}\)-operations are
\[ (\omega_\nu)R^c_{Y;\mathfrak m^c} \]
-operations, whose bases are the bases of the \(\omega_\nu\)-disjunctive extension of \(\Delta\Sigma\)-operations with bases of the family
\[ \mathfrak M^c=(N_Y^c)_{Y\in W}. \]
They can also be obtained by means of the \(R^c\)-process, starting from operations with bases of the family \(\mathfrak M^c\).

For an \((\omega_\nu)R_{Y;\mathfrak m}\)-operation one may define external indices:
\[ (\omega_\nu)R_{Y;\mathfrak m}\operatorname{Ind}(x\mid\{E_Y\}) = \begin{cases} \omega_{\nu+1}, & \text{if } x\in E_Y^{\omega_{\nu+1}},\\ \beta, & \text{if } x\in\displaystyle\bigcap_{\alpha<\beta}E_Y^\alpha\setminus E_Y^\beta\quad(\beta<\omega_{\nu+1}). \end{cases} \]

The internal index of an \((\omega_\nu)R_{Y;\mathfrak m}\)-operation is defined at points \(x\in E_Y^{\omega_{\nu+1}}\). It is equal to the least of the numbers \(\beta\) such that the structure of the family of sets \((E_Y^\alpha)_{Y\in W}\) remains unchanged for all \(\alpha\geqslant\beta\).

The external indices \((\omega_\nu)R_{\mathfrak M}\)-operations satisfy the principle of comparison of indices:

Let \(\mathfrak M_1=(N_Y)_{Y\in W}\), \(\mathfrak M_2=(P_Y)_{Y\in W}\) be families of bases, where \(N_Y\subseteq \Xi\), \(P_Y\subseteq \Xi\); \((E_Y)_{Y\in W}\), \((H_Y)_{Y\in W}\) are arbitrary families of sets; \((E_Y^\alpha)_{Y\in W,\alpha<\omega_{\nu+1}}\), \((H_Y^\alpha)_{Y\in W,\alpha<\omega_{\nu+1}}\) are the derived sets of these families;

\[ \Phi_{M_Y}^{\alpha+1}\equiv R_{Y;\{N_Y^\alpha\}};\qquad \Phi_{Q_Y}^{\alpha+1}\equiv R_{Y;\{P_Y^\alpha\}} \quad\text{for }\alpha<\omega_\nu;\qquad U_{YY'}=E_Y\cup C H_{Y'}, \]

\[ \Phi_{L_{YY'} }^{\alpha+1}\{U_{YY'}\} = \Phi_{Y'_1,Q_{Y'} }^{-\alpha+1,c} \{\Phi_{Y_1,M_Y}^{-\alpha+1}\{U_{Y_1Y'_1}\}\}, \]

where

\[ \Phi_{\overline{M_Y}^{\alpha+1}}\{E_{Y'}\} = E_Y\cap \Phi_{M_Y}^{\alpha+1}\{E_{Y'}\}, \qquad \Phi_{\overline{Q_Y}^{\alpha+1}}\{E_{Y'}\} = E_Y\cap \Phi_{Q_Y}^{\alpha+1}\{E_{Y'}\}. \]

Then the \(R\)-iteration with variable bases
\[ \mathfrak L=(L_{YY'}^{\alpha+1})_{Y,Y'\in W,\alpha<\omega_{\nu+1}}, \]
performed over the family of sets \((U_{YY'})\), gives, for any \(\alpha<\omega_{\nu+1}\),

\[ U_{YY'}^\alpha=\bigcap_{\beta\leq\alpha}(E_Y^\beta\cup C H_{Y'}^\beta), \]

\[ (\omega_\nu)R_{\mathfrak D}\{U_{YY'}\} = \{x\mid(\omega_\nu)R_{\mathfrak M_1}\operatorname{Ind}(x\mid\{E_Y\}) \geq (\omega_\nu)R_{\mathfrak M_2}\operatorname{Ind}(x\mid\{H_Y\})\}. \]

Let \(\mathfrak M=(N_\alpha)_{\alpha\in I\cup W_{\mathfrak M}}\), where \(N_\alpha\subseteq\Xi\), and let \(W_{\mathfrak M}\) be the set of all coordinated sequences \((i_\alpha)_\gamma=Y_\gamma\), for \(\gamma\in K_{\mathrm{II}}\), \(\gamma<\omega_\nu\), in which \(i_{\alpha+1}\), \(i_\chi\) for \(\chi\in K_{\mathrm{II}}\) are immediately subordinate, respectively, to \(i_\alpha,(i_\alpha)_\chi\). An \((\omega_\nu)T_{\mathfrak M}\)-operation over a family of sets \((E_i)_{i\in I}\) is called a \(\Delta\Sigma\)-operation whose base \(M\) is the base of an \(\omega_\nu\)-conjunctive extension of \(\Delta\Sigma\)-operations with bases of the family \(\mathfrak M\), i.e.

\[ \mu\in M\equiv 0\in\mu\ \&\ (\forall i\in\mu)(\exists\eta\in N_i)[\eta\subset\mu]\ \& \]

\[ \&\ (\forall (i_\alpha)_\gamma\in W_{\mathfrak M}) [(i_\alpha)_\gamma\subset\mu\to (\exists\eta\in N_{(i_\alpha)_\gamma})[\eta\subset\mu]]. \]

The base of the complementary operation \((\omega_\nu)T_{\mathfrak M^c}^{c}\) is the base of an \(\omega_\nu\)-disjunctive extension of \(\Delta\Sigma\)-operations with bases of the family
\[ \mathfrak M^c=(N_\alpha^c)_{\alpha\in I\cup W_{\mathfrak M}}. \]

The operations \((\omega_\nu)T_{\mathfrak M}\) and \((\omega_\nu)T_{\mathfrak M^c}^{c}\) are called operations without interlocking if there exists a family of pairwise nonintersecting chains \((\eta^\alpha)_{\alpha\in I\cup W_{\mathfrak M}}\) having the properties:

\[ (\mathrm H)\ \forall\alpha\,[\eta^\alpha\sim I\ \&\ 0\notin\eta^\alpha\ \&\ (\alpha\in I\to \alpha\notin\eta^\alpha)\ \&\ (\alpha\in W_{\mathfrak M}\to(\forall i\in\alpha)[i\notin\eta^\alpha])]. \]

\[ (\mathrm H_{\mathfrak M}^{c})\ \forall\alpha\ (\forall\eta\in N_\alpha)\ [\eta\subseteq\eta^\alpha]. \]

Every \((\omega_\nu)T\)-operation without interlocking is equivalent to some \((\omega_\nu)R\)-operation, and conversely. Let the base \(N\) satisfy the conditions

\[ 1^\circ)\quad (\omega_\nu)T_N\succ \bigcup_\tau,\quad (\omega_\nu)T_N\succ \bigcap_\tau,\quad (\omega_\nu)T_N\succ \Phi_{N^c},\quad (\omega_\nu)T_N\succ \Phi_{N\alpha c} \quad\text{for }\alpha<\omega_\nu, \]

\[ [1^{00})\quad (\omega_\nu)T_N\succ \bigcup_\tau,\quad (\omega_\nu)T_N\succ \bigcap_\tau,\quad (\omega_\nu)T_N\succ \Phi_{N^c}]. \]

\[ 2^\circ)\quad (\Phi_N,d)\prec \Phi_N,\quad (\Phi_{N^c},d)\prec \Phi_{N^c}. \]

Condition \(2^\circ\) may be replaced by the condition

\[ 2')\quad (\omega_\nu)T_{T_N}\prec(\omega_\nu)T_N,\qquad ((\omega_\nu)T_N^c,d)\prec(\omega_\nu)T_N^c. \]

Denote by \((\omega_\nu)\mathfrak B_N\) the class of \(\Delta\Sigma\)-operations possessing the following properties: 1) the trivial operations \(\bigcap,\bigcup,\Phi_N,\Phi_{N^c},\Phi_N^\alpha,\Phi_N^{\alpha c},T_{\{N^\alpha\}},T_{\{N^{\alpha c}\}}^{c}\), for all \(\alpha<\omega_\nu\), belong to the class \((\omega_\nu)\mathfrak B_N\); 2) if \(\Phi_L\in(\omega_\nu)\mathfrak B_N\), \(\Phi_{M_i}\in(\omega_\nu)\mathfrak B_N\) for \(i\in I\), then \(\Phi_L\{\Phi_{M_i}\}\in(\omega_\nu)\mathfrak B_N\); 3) the class \((\omega_\nu)\mathfrak B_N\) is closed with respect to shifts of bases; 4) \((\omega_\nu)\mathfrak B_N\) is the smallest class of sets satisfying conditions 1)—3). By \((\omega_\nu)T\mathfrak B_N\) we denote the class of \(\omega_\nu\)-conjunctive extensions of \(\Delta\Sigma\)-operations of the class \((\omega_\nu)\mathfrak B_N\). Under the given conditions, the operations belonging to the classes \((\omega_\nu)\mathfrak B_N\) and \((\omega_\nu)T\mathfrak B_N\) are no stronger than operations of type \((\omega_\nu)T_N\).

For each \((\omega_\nu)T\)-operation one can construct equivalent operations leading to an increase of the index of the initial operation by one and to a doubling of the index.

Let \(N \subseteq \Xi\). Put \(\Phi_{N_0} \equiv \Phi_N\). The type of a \(\Delta\Sigma\)-operation \((\omega_\nu)T_N=\Phi_N\) will be denoted by \((\omega_\nu)T_N^1\). If the type \((\omega_\nu)T_N^\alpha\) is defined, then by \((\omega_\nu)T_N^{\bar{\alpha}}\) we denote the type of supplementary operations to operations of type \((\omega_\nu)T_N^\alpha\). The type
\[ (\omega_\nu)T_N^{\alpha+1}\equiv(\omega_\nu)T_{\{N_\alpha^c\}}\equiv\Phi_{N_{\alpha+1}}, \]
where
\[ \Phi_{N_\alpha^c}\equiv(\omega_\nu)T_N^\alpha . \]
If \(\Delta\Sigma\)-operations \(\Phi_{N_\alpha}\) are defined for all \(\alpha<\varkappa<\omega_{\nu+1}\), \(\varkappa\in K_{\mathrm{II}}\), then let \((\beta_j)\to\varkappa\), \(\beta_j<\varkappa\);
\[ I=\bigcup_j \eta^j \]
be a decomposition of the space \(I\) such that \(\forall j[\eta^j\sim I]\);
\[ \varphi_j(t)=t:I\to\eta^j \]
is a bijection. Then
\[ \Phi_{N_\varkappa}\{E_i\}= \bigcap_{(\beta_j)\to\varkappa} \Phi_{N_{\varphi_j}}\{E_{\varphi_j(i)}\}. \]

The type of \(\Delta\Sigma\)-operation
\[ \Phi_{N_{\varkappa+1}}\equiv(\omega_\nu)T_{\{N_\varkappa\}} \]
will be denoted by \((\omega_\nu)T_N^{\varkappa+1}\). The strengthening of types of \(\Delta\Sigma\)-operations can also be continued to transfinite numbers \(\alpha\), where \(\omega_{\nu+1}<\alpha<\omega_{\nu+2}\), as was done in work \((^2)\), Chapter IV. If the constructed operations are applied to some class of sets \(K\), then we obtain an \((\omega_\nu)R_N\)- or \((\omega_\nu)T_N\)-classification of sets. This classification is monotone if the base \(N\) satisfies condition \(1^0\). If the base \(N\) satisfies conditions \(1^0\) and \(2^0\), then the operations \((\omega_\nu)T_N^{\alpha+1}\) are normal,
\[ (\omega_\nu)T_{N_\alpha^c}\equiv(\omega_\nu)T_{N_\alpha^c} \]
with respect to the class of sets \(K\supset\varnothing,\mathfrak{N}\). If \(N\) satisfies condition \(1^0\), then each of the bases \(N_{\alpha+1}^c,N_\varkappa\), for \(\varkappa\in K_{\mathrm{II}}\), satisfies this condition.

If the class of sets \(K\supset\varnothing,\mathfrak{N}\) is closed with respect to the operations of complementation and \(\bigcup_n\) for \(n<\omega\), and the base \(N\) satisfies conditions \(1^0,2^0\) or \(2'\), then the class of external indices of operations of type \((\omega_\nu)T_N^{\alpha+1}\) for \(\alpha<\omega_{\nu+1}\) is completely regular, and for the classes \((\omega_\nu)T_N^{\alpha+1}(K)\) the first and second separation laws and the multiple separation law hold with respect to operations \(\Phi_{\mathfrak{M}}\) such that the class \((\omega_\nu)T_N^{\alpha+1}(K)\) is closed with respect to the operations \(\Phi_{\mathfrak{M}i}\) for \(i\in I\). These include the operations \((\omega_\nu)T_N^\beta\) for \(\beta\le\alpha+1\), \((\omega_\nu)T_{N^{\beta c}}\) for \(\beta\le\alpha\).

If the base \(N\) satisfies conditions \(1^0,2^0\) or \(2'\) and condition \(3^0\). If \(L\subseteq\mathfrak{M}_{\tau^*}\), where
\[ \mathfrak{M}=\{N\}\cup\{N^i\}_{i\in I},\qquad \tau^*\le\aleph_{\nu+1}, \]
and \(\mathfrak{M}_{\tau^*}\) is the totality of intersections of \(<\tau^*\) sets of the family \(\mathfrak{M}\), then
\[ \Phi_L<\Phi_N^* \]
or
\[ \Phi_L<T_N; \]
then for the class \((\omega_\nu)T_N^{\varkappa+1}(K)\), \(K\supset\varnothing,\mathfrak{N}\), and the properties \(H_p\) for \(2\le p<\omega\), the covering theorems and expressions 4, 5 of work \((^3)\) hold.

We carry out the realization of the \((\omega_\nu)R_N\)-classification of sets in the Baire space \(J^\omega\), starting from the operation
\[ \Phi_N\equiv\bigcup_\tau \]
and the class of open-closed sets of this space. We obtain the classes of \((\omega_\nu)R\)-sets
\[ (\omega_\nu)R_\alpha,\quad (\omega_\nu)CR_\alpha,\quad (\omega_\nu)BR_\alpha=(\omega_\nu)R_\alpha\cap(\omega_\nu)CR_\alpha . \]
Each of the classes \((\omega_\nu)R_\alpha,\;(\omega_\nu)CR_\alpha\) has a universal set, is topologically invariant, and the classification of sets is essentially monotone. The question of the coincidence of the class
\[ (B_1)=(\omega_\nu)BR_1 \]
with the class of \(B\)-sets of this space remains open; here \((B)\subset(B_1)\). The question of the approximability of sets is open. The base \(N\) of the operation \(\bigcup_\tau\) satisfies conditions \(1^0,2',3^0\).

Complete bases of operations
\[ (\omega_\nu)T^\alpha\equiv(\omega')T_N^\alpha \]
for
\[ \Phi_N\equiv\bigcup_\tau \]
belong to the class \((B_2)\) of projective sets of the space \(J^{\omega_\nu}\). The class of \((\omega_\nu)R\)-sets also belongs to the class \((B_2)\).

Volgograd Pedagogical Institute
named after A. S. Serafimovich

Received
30 XII 1969

CITED LITERATURE

1. A. A. Lyapunov, Algebra and Logic, Seminar, 2, 2, 47 (1963).
2. A. A. Lyapunov, Proceedings of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 40, 3 (1953).
3. Z. I. Kozlova, Doklady Akademii Nauk, 188, No. 5, 1001 (1969).