Full Text
UDC 519.52
MATHEMATICS
Z. I. KOZLOVA
PROJECTIVE SETS OF TOPOLOGICAL SPACES OF WEIGHT \(\tau\)
(Presented by Academician P. S. Novikov on 6 III 1968)
Let \(I\) be a space of indices of cardinality \(\tau=\aleph_\nu\), which we shall regard as a strongly inaccessible cardinal number \((^1)\). We study the projective hierarchies of classes of sets of topological spaces \(D^\tau, J^\tau\) and topological \(\tau\)-spaces \(D^{\omega_\nu}, J^{\omega_\nu}\) \((^3)\). We shall show that the classes of these hierarchies are determined by substantially different operations, depending on the topology of the space.
The operation
\[
\Psi^{x}_{N(x)}\{E_i\}=\bigcup_x \bigl(\{x\}\cap \Psi^\circ_{N(x)}\{E_i\}\bigr)
\]
is called a set-theoretic operation with variable base. A class \(P\subset \mathfrak{P}\mathfrak{R}_{xy}=\mathfrak{P}(\mathfrak{R}_x\times \mathfrak{R}_y)\), where \(\mathfrak{R}\) is the basic space, is called projective with respect to the class \(\mathfrak{K}\subset \mathfrak{P}\mathfrak{R}_x\) if there exists such a variable base \(L(y)\) that
\[
P=\Psi^{(x,y)}_{L(y)}(\mathfrak{K}^*),
\]
where \(\mathfrak{K}^*\) is the class of all sets of the form \(K\times \mathfrak{R}_y\) \((K\in \mathfrak{K})\),
\[
\Psi^{(x,y)}_{L(y)}\{E_i\}=\Psi^{(x,y)}_{L(x,y)}\{E_i\},
\quad \text{when } L(x,y)=L(x) \quad ({}^2).
\]
L. V. Kantorovich and E. M. Livenson \((^2)\) showed that if a class \(P\) is projective with respect to a class \(\mathfrak{K}\), then for the class \(P'\) of projections of sets belonging to \(P\) the equality
\[
P'=\Psi_{L_0}(\mathfrak{K})
\]
holds, where
\[
L_0=\bigcup_{y\in \mathfrak{R}_y} L(y).
\]
Theorem 1. The classes \(F\) and \(G\) of the space \(J^{\omega_\nu}_{xy}\) \((D^{\omega_\nu}_{xy}, D^\tau_{xy}, J^\tau_{xy})\) are projective, respectively, with respect to the classes \(F\) and \(G\) of the space \(J^{\omega_\nu}_{x}\) \((D^{\omega_\nu}_{x}, D^\tau_x, J^\tau_x)\). The base \(L(y)\) for the class \(G\subset \mathfrak{P}J^{\omega_\nu}_{xy}\), where \(y=\{i_0,\ldots,i_\alpha,\ldots\}\in J^{\omega_\nu}_{xy}\), consists of all chains of the family \((\{\lambda(i_0,\ldots,i_\alpha)\})_\alpha\); the base \(L^c(y)\) consists of one chain
\[
\{\lambda(i_0),\lambda(i_0,i_1),\ldots,\lambda(i_0,i_1,\ldots,i_\alpha),\ldots\},
\]
where \(\lambda(i_0,\ldots,i_\alpha)=\lambda\) is a one-to-one mapping of the space of all tuples \(W\) of the space of indices \(I\) onto \(I\). The base \(L_1(y)\) for the class \(G\subset \mathfrak{P}J^\tau_{xy}\) consists of all chains of the family
\[
(\{\lambda(i_{\alpha_1},\ldots,i_{\alpha_k})\})_{\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega},
\]
where \(I''_\omega\) is the totality of all finite tuples \(\{\alpha_1,\ldots,\alpha_k\}\) of the space of indices \(I\), \(\alpha_1<\alpha_2<\cdots<\alpha_k\), and \(\lambda(i_{\alpha_1},\ldots,i_{\alpha_k})=\lambda\) is a one-to-one mapping of the space \(X^*\) of all tuples \(\{i_{\alpha_1},\ldots,i_{\alpha_k}\}\), for \(\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega\), onto \(I\). The base \(L^c_1(y)\) consists of one chain
\[
(\lambda(i_{\alpha_1},\ldots,i_{\alpha_k}))_{\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega}.
\]
The base \(L'(y)\) for the class \(G\subset \mathfrak{P}D^\tau_{xy}\), where \(y=\{\nu_0,\ldots,\nu_\alpha,\ldots\}\in D^\tau_y\), consists of chains of the family
\[
(\{\lambda(\nu_{\alpha_1},\ldots,\nu_{\alpha_k})\})_{\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega};
\]
the base \(L^c_1(y)\) for the class \(F\subset D^\tau_{xy}\) consists of one chain
\[
(\lambda(\nu_{\alpha_1},\ldots,\nu_{\alpha_k}))_{\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega}.
\]
The base \(L''(y)\) for the class \(G\subset \mathfrak{P}D^{\omega_\nu}_{xy}\), where \(y=\{\nu_0,\ldots,\nu_\alpha,\ldots\}\in D^{\omega_\nu}_y\), consists of chains of the family
\[
(\{\lambda(\nu_0,\ldots,\nu_\alpha)\})_\alpha.
\]
The base \(L''{}^c(y)\) for the class \(F\subset \mathfrak{P}D^{\omega_\nu}_{xy}\) consists of one chain
\[
(\lambda(\nu_0,\ldots,\nu_\alpha))_\alpha.
\]
Corollary 1. The classes of projections of sets belonging to the classes \(\Phi_M(F)\) and \(\Phi_M(G)\) of the spaces \(D^\tau_{xy}, D^{\omega_\nu}_{xy}, J^{\omega_\nu}_{xy}, J^\tau_{xy}\) are respectively the \(\Delta\Sigma\)-classes \(\Phi_{M'_1}(F)\) and \(\Phi_{M''_2}(G)\), \(\Phi_{M'_1}(F)\) and \(\Phi_{M''_2}(G)\), \(\Phi_{M_1}(F)\) and \(\Phi_{M_2}(G)\), \(\Phi_{11'}(F)\) and \(\Phi_{M_{12}}(G)\), which may be represented respectively by the operations
\[
\bigcup_{\nu_0,\ldots,\nu_\alpha,\ldots}\ \Phi_{M'_i}\left\{
\bigcap_{\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega}
E_{j;\,\nu_{\alpha_1}\ldots\nu_{\alpha_k}}
\right\}
=
\Phi_{\tilde S',\tilde M}\{E_j\},
\qquad
E_{j;\,\nu_{\alpha_1}\ldots\nu_{\alpha_k}}\in F\subset \mathfrak{P}D^\tau,
\]
\[
\bigcup_{\nu_0,\ldots,\nu_a}\Phi_M\left\{\bigcup_j\bigcup_{\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega}
E_j;\nu_{\alpha_1}\ldots\nu_{\alpha_k}\right\}
=\Phi_{\underline{S}''\widetilde M}\{E_{j\lambda}\},\qquad
E_j;\nu_{\alpha_1}\ldots\nu_{\alpha_k}\in G\subset\mathfrak P D^\tau,
\]
where \(\lambda=\lambda(\nu_{\alpha_1},\ldots,\nu_{\alpha_k})\);
\[ \bigcup_{\alpha_3,\ldots,\nu_\alpha,\ldots}\Phi_M\left\{\bigcup_j\bigcap_\alpha E_j;\nu_0\ldots\nu_\alpha\right\} =\Phi_{\mathfrak B\widetilde M}\{E_{j\lambda}\},\qquad E_j;\nu_3\ldots\nu_\alpha\in F\subset\mathfrak P D^{\omega_\nu}, \]
\[
\bigcup_{\alpha_0,\ldots,\nu_\alpha,\ldots}\Phi_M\left\{\bigcup_j\bigcup_\alpha
E_j;\nu_3\ldots\nu_\alpha\right\}
=\Phi_{\mathfrak B\widetilde M}\{E_{j\lambda}\},\qquad
E_j;\nu_0\ldots\nu_\alpha\in G\subset\mathfrak P D^{\omega_\nu},
\]
where \(\lambda=\lambda(\nu_0,\ldots,\nu_\alpha)\);
\[ \bigcup_{i_3,\ldots,i_\alpha,\ldots}\Phi_M\left\{\bigcup_j\bigcap_\alpha E_j;i_3\ldots i_\alpha\right\} =\Phi_{\mathfrak A\widetilde M}\{E_{j\lambda}\},\qquad E_j;i_3\ldots i_\alpha\in F\subset\mathfrak P J^{\omega_\nu}, \]
\[
\bigcup_{i_3,\ldots,i_\alpha,\ldots}\Phi_M\left\{\bigcup_j\bigcup_\alpha
E_j;i_3\ldots i_\alpha\right\}
=\Phi_{\mathfrak A\widetilde M}\{E_{j\lambda}\},\qquad
E_j;i_3\ldots i_\alpha\in G\subset\mathfrak P J^{\omega_\nu},
\]
where \(\lambda=\lambda(i_0,\ldots,i_\alpha)\);
\[ \bigcup_{i_0,\ldots,i_\alpha,\ldots}\Phi_M\left\{\bigcap_j \bigcup_{\{\alpha_1,\ldots,\alpha_k\}\in I'_\omega} E_j;i_{\alpha_1}\ldots i_{\alpha_k}\right\} =\Phi_{\underline S\widetilde M}\{E_{j\lambda}\},\qquad E_j;i_{\alpha_1}\ldots i_{\alpha_k}\in F\subset\mathfrak P J^\tau, \]
\[
\bigcup_{i_0,\ldots,i_\alpha,\ldots}\Phi_M\left\{\bigcup_j
\bigcup_{\{\alpha_1,\ldots,\alpha_k\}\in I''_\omega}
E_j;i_{\alpha_1}\ldots i_{\alpha_k}\right\}
=\Phi_{\breve S\widetilde M}\{E_{j\lambda}\},\qquad
E_j;i_{\alpha_1}\ldots i_{\alpha_k}\in G\subset\mathfrak P J^\tau,
\]
where \(\lambda=\lambda(i_{\alpha_1},\ldots,i_{\alpha_k})\).
Corollary 2. The class of projections of sets belonging to the class \(G\) of the spaces under study is the class \(G\). The class of projections of sets belonging to the class \(F\) in the spaces \(D^\tau, D^{\omega_\nu}\), is contained in the class of \((\omega_\nu)\) \(A\)-sets in the space \(J^{\omega_\nu}\), and belongs to the class \(\Phi_S(F)\) in the space \(J^\tau\).
Theorem 2. The bases of the \(\Delta\Sigma\)-operations \(M_1', M_2', M_1'', M_2'', M_1, M_2, M_{11}, M_{12}\) of Corollary 1 can be represented in the form
\[
M_1'=\Phi_M\{K_s'\},\quad M_2'=\Phi_M\{V_s'\};\quad
M_1''=\Phi_M\{K_s''\},\quad M_2''=\Phi_M\{V_s''\};
\]
\[
M_1=\Phi_M\{K_s\},\quad M_2=\Phi_M\{V_s\};\quad
M_{11}=\Phi_M\{K_s^1\},\quad M_{12}=\Phi_M\{V_s^1\},
\]
where \(K_s', V_s', K_s'', V_s'', K_s, V_s, K_s^1, V_s^1\) are certain sets of type \(F_\Sigma\) in \(J^{\omega_\nu}\) (independent of \(M\)).
The operation \(\Phi_{\mathfrak A}\) is incomparable with the operations \(\Phi_{\mathfrak B}, \Phi_{S''}\), and at the same time the operation \(\Phi_{\mathfrak A}\) is stronger than the operation \(\Phi_{\mathfrak B}\) with respect to any class of sets \(K\supset\varnothing\), and essentially stronger than \(\Phi_{S''}\) and \(\Phi_S\) with respect to any class of sets invariant with respect to the operation \(\bigcap_\tau\). In view of this, the composition \(\Phi_{\mathfrak A\widetilde M}\) is stronger than the composition \(\Phi_{\mathfrak B\widetilde M}\) with respect to the class of sets \(K\supset\varnothing\), and stronger than \(\Phi_{S''\widetilde M}\) and \(\Phi_{\breve S\widetilde M}\) with respect to any class of sets invariant with respect to the operation \(\bigcap_\tau\). Consequently, projective operations substantially depend on the topology of the space. Since in the spaces under study the class \(F\supset\varnothing\) and is invariant with respect to the operation \(\bigcap_\tau\), we have \(\Phi_{\mathfrak A\widetilde M}(F)\supset\Phi_{S''\widetilde M}(F)\), \(\Phi_{\mathfrak A\widetilde M}(F)\supset\Phi_{\mathfrak B\widetilde M}(F)\), \(\Phi_{\mathfrak A\widetilde M}(F)\supset\Phi_{\breve S\widetilde M}(F)\) for any base \(M\).
The classes of the compositional hierarchies of the space \(J_{xy}^{\omega_\nu}\) are constructed as follows. Let \(N=\mathfrak A\), where \(\mathfrak A\) is a rigid base of an \(A\)-operation with a complete depth chain, \(\Phi_M\equiv\bigcup_\tau\). The operation \(\Phi_{\mathfrak A}\) and the composition \(\Phi_{\mathfrak A\widetilde{\mathfrak A}}\) are normal, and the operation \(\Phi_M\) satisfies the condition:
\[
1^*. \ \Phi_M\prec\Phi_{\mathfrak A},\ \Phi_{Mc}\prec\Phi_{\mathfrak A}.
\]
The \(\Phi_{\mathfrak A M}\)-hierarchy of classes of sets generated by the class \(K_0\) of open-closed sets of the space \(J^{\omega_\nu}\) forms the classes of projective sets of this space:
\[
P_0=\Phi_{\mathfrak A M}^0(K_0)=\Phi_M(K_0)=\Phi_L(K_0)=G,
\]
\[ CP_0=\Phi_{\mathfrak A M}^{0c}(K_0)=\Phi_M^c(K_0)=\Phi_{L_0^c}(K_0)=F, \]
\[ P_{\alpha+1}=\Phi_{\mathfrak A M}^{\alpha+1}(K_0)=\Phi_{\mathfrak A\,\breve L_\alpha^c}(K_0)=\Phi_{L_{\alpha+1}}(K_0), \]
\[ CP_{\alpha+1}=\Phi_{\mathfrak A M}^{\alpha+1\,c}(K_0)=\Phi_{\mathfrak A\,\breve L_\alpha}(K_0)=\Phi_{L_{\alpha+1}^c}(K_0), \]
\[ P_\chi=\Phi_{\mathfrak A M}^{\chi}(K_0)=\bigcup_{(\alpha i)\to\chi}\Phi_{\mathfrak A M}^{\alpha_i}(K_0)=\Phi_{L_\chi}(K_0), \]
\[ CP_\chi=\Phi_{\mathfrak A M}^{\chi c}(K_0)=\bigcup_{(\alpha i)\to\chi}\Phi_{\mathfrak A M}^{\alpha_i c}(K_0)=\Phi_{L_\chi^c}(K_0), \]
where \(\chi<\omega_{\nu+1}\) is a limit transfinite number, \((B_\alpha)=BP_\alpha=P_\alpha\cap CP_\alpha\).
Properties of the classes of projective sets:
-
\(P_1=(A)\), \(CP_1=(CA)\), \(B_1=(A)\cap(CA)=(B)\), where \((B)\) is the class of \(B\)-sets of the space \(J^{\omega_\nu}\).
-
\(P_\alpha\subset CP_{\alpha+1}\), \(CP_\alpha\subset P_{\alpha+1}\), \(P_\alpha\subset P_{\alpha+1}\) for every \(\alpha<\omega_{\nu+1}\).
-
The classes \(P_\alpha\) for \(\alpha<\omega_{\nu+1}\) are invariant with respect to the operation of projection. The classes \(P_{\alpha+1}\) and \(CP_{\alpha+1}\) are invariant with respect to the operations \(\bigcup_\tau\) and \(\bigcap_\tau\). The class \(P_\chi\) is invariant with respect to the operations \(\bigcup_\tau\) and intersection of finite families of sets.
-
Under a homeomorphic transformation of the space \(J^{\omega_\nu}\) onto itself, sets of the class \(P_\alpha(CP_\alpha)\) pass into sets of the same class \((\alpha<\omega_{\nu+1})\).
-
If \(N\in P_{\alpha+1}\), \(E_i\in P_{\alpha+1}\) for every \(\alpha<\omega_{\nu+1}\), then \(E=\Phi_N\{E_i\}\in P_{\alpha+1}\).
In the space \(J^{\omega_\nu}\) the class of projections of \(B\)-sets coincides with the class of \(A\)-sets of this space, which we shall denote by \((A_1)\). If, in accordance with the projective hierarchy of classes of sets of this space, the projective class \((A_\alpha)\) is defined, then the class of complements to sets of the class \((A_\alpha)\) we shall denote by \((CA_\alpha)\). The class of projections of sets of the class \((CA_\alpha)\) we denote by \((A_{\alpha+1})\). If \(\chi<\omega_{\nu+1}\) is a limit transfinite number, then by the class \((A_\chi)\) we mean the class of unions of projective sets of classes \(<\chi\) of any family of cardinality \(\leq\tau\).
Theorem 3. The classes of projective sets \((A_\alpha)\) and \((CA_\alpha)\) for \(0<\alpha<\omega_{\nu+1}\) can be obtained by means of the operations \(\Phi_{S_{\alpha+1}^{(i)}}\), for \(i=1,2,3,4\), starting from the classes of sets \(F\) and \(G\):
\[ (A_\alpha)=\Phi_{S_\alpha^{(1)}}(F)=\Phi_{S_\alpha^{(2)}}(G)=A^\alpha(F); \]
\[ (CA_\alpha)=\Phi_{S_\alpha^{(3)}}(F)=\Phi_{S_\alpha^{(4)}}(G)=CA^\alpha(F), \]
where \(S_\alpha^{(1)}, S_\alpha^{(2)}\in(CA_{\alpha-1})\), if \(\alpha\) is a nonlimit ordinal number, and \(S_\alpha^{(1)}, S_\alpha^{(2)}\in(A_\alpha)\), if \(\alpha\) is a limit ordinal number; \(S_\alpha^{(3)}, S_\alpha^{(4)}\in(CA_\alpha)\) for \(\alpha<\omega_{\nu+1}\).
Theorem 4. Each projective operation \(A^\alpha\) and \(CA^\alpha\) for \(0<\alpha<\omega_{\nu+1}\) can be given by a \(\Delta\Sigma\)-operation with a rigid base.
Theorem 5. The operations \(\Phi_{S_{\alpha+1}^{(i)}}\) for \(i=1,2\) are normal with respect to any class \((A_\beta)\) and with respect to classes of sets \((CA_\beta)\) for \(\beta\leq\alpha\), while the operations \(\Phi_{S_{\alpha+1}^{(i)}}\) for \(i=3,4\) are normal with respect to any class \((CA_\beta)\) and with respect to classes \((A_\beta)\) for \(\beta\leq\alpha\).
Theorem 6. The operations \(\Phi_{L_\alpha}\) and \(A^\alpha\), \(\Phi_{L_\alpha^c}\) and \(CA^\alpha\) are equivalent for \(\alpha<\omega_{\nu+1}\).
Corollary 1. \(P_\alpha=(A_\alpha)\), \(CP_\alpha=(CA_\alpha)\) for every \(\alpha<\omega_{\nu+1}\).
Corollary 2. In the space \(J^{\omega_\nu}\), the class of \(C\)-sets belongs to the class \((\overline{B}_2)\).
Corollary 3. Complete bases of all \((\omega_\nu)R^\alpha\)-operations belong to the class \((\overline{B}_2)\). The class of \((\omega_\nu)R\)-sets of the space \(J^{\omega_\nu}\) is contained in the class \((B_2)\).
Corollary 4. If the class \(P_{\alpha+1}\) is invariant with respect to all operations with bases of the family \(\mathfrak{M}=(M_i)\cup(M_{(i\alpha)_\gamma})\), where \(i\in I\), \((i\alpha)_\gamma\) is the totality of all concordant sequences of limit type \(\gamma\leqslant\omega_\nu\), then it is also invariant with respect to the operation \((\omega_\nu)T\).
It follows from this that the operation of type \((\omega_\nu)T_{\{L_{\alpha+1}\}}\) is equivalent to the operation \(\Phi_{L_{\alpha+1}}\) with respect to any class of projective sets for any \(\alpha<\omega_{\nu+1}\); the operation of type \((\omega_\nu)P_{\mathfrak{A}\check{\mathfrak{M}}^{c}}\), where \(\Phi_M\equiv\bigcup_\tau\), is equivalent to the operation \(\Phi_{\check{\mathfrak{A}}L_1^c}\equiv\Phi_{L_2}\); the operation \((\omega_\nu)P_{\check{\mathfrak{A}}L_{\alpha+1}^{c}}\) is equivalent to the operation \(\Phi_{L_{\alpha+2}}\) with respect to any class of projective sets for \(0<\alpha<\omega_{\nu+1}\).
When the generalized continuum axiom is added, for the class of sets \(P_2\) the separation laws that hold are the same as for the second class of projective sets of the space \(J\).
Let the base be \(N=S\), \(\Phi_M\equiv\bigcup_\tau\). The operation \(\Phi_S\) and the composition \(\Phi_{\check S}\check S\) are normal (the latter with respect to the class \(K\supset J^\tau\)), and the operation \(\Phi_M\) satisfies condition \(1^*\). The \(\Phi_{SM}\)-hierarchy of classes of sets generated by the class \(K_0\) of open-closed sets of the space \(J^\tau\) forms classes of projective sets of this space, which we shall denote in the same way as for the space \(J^{\omega_\nu}\). The class \(P_0=G\), \(CP_0=F\), \((B_0)=K_0\), \(P_1\subset(A)\), \(CP_1\subset(CA)\), \((B_1)\subset(B)\). The class of projections of \(B\)-sets of the space \(J^\tau\) coincides with the class of sets \(P_1\). The class of \((\omega_\nu)A\)-sets of this space contains, as a proper part, the class of projections of \(B\)-sets of the given space.
Let the base be \(N=S''\); the operation \(\Phi_M\) is equivalent to the operation \(\Phi_{M_1}\{E_{ij}\}=\bigcup_i\bigcap_j E_{ij}\). The operation \(\Phi_{S''}\) is nonnormal, but the composition \(\Phi_{\check S''S''}\) is normal with respect to the class \(K\supset D^\tau\). The operation \(\Phi_M\) does not satisfy condition \(1^*\). The \(\Phi_{S''\check M}\)-hierarchy of classes of sets, generated by the class \(K_0\) of open-closed sets of the space \(D^\tau\), forms a hierarchy of projective classes of the space \(D^\tau\). In this case the class \(P_0=G^2\), \(CP_0=F^2\), \((B_0)=B^2\). The classes of projective sets are monotone. For even \(\alpha\) the class \(P_\alpha\) is invariant with respect to the operation \(\bigcup_\tau\), and the class \(P_{\alpha+1}\) with respect to the operation \(\bigcap_\tau\). The class \(P_1\subset(A)\) in the space \(D^\tau\).
Let the base be \(N=\mathfrak{B}\); \(\Phi_M\) is equivalent to the operation \(\Phi_{M_1}\{E_{ij}\}=\bigcup_i\bigcap_i E_{ij}\). The operation \(\Phi_{\mathfrak{B}}\) is nonnormal, but the composition \(\Phi_{\check{\mathfrak{B}}\mathfrak{B}}\) is normal with respect to the class of sets \(K\supset D^{\omega_\nu}\). The operation \(\Phi_M\) does not satisfy condition \(1^*\). The \(\Phi_{\mathfrak{B}\check M}\)-hierarchy of classes of sets forms a projective hierarchy of the space \(D^{\omega_\nu}\). The properties of the classes of sets \(P_\alpha\) of the space \(D^{\omega_\nu}\) coincide with the properties of the corresponding classes of the space \(D^\tau\). \(P_1\subset(A)\) in the space \(D^{\omega_\nu}\).
Volgograd Pedagogical Institute
named after A. S. Serafimovich
Received
6 II 1968
REFERENCES
- N. Bourbaki. Set Theory, Moscow, 1965.
- L. V. Kantorovich, E. M. Livenson, Fundamenta Math., 18, 214 (1932).
- I. Kozlova, Scientific Notes of the Volgograd Pedagogical Institute (1968).