UDC 519.52

MATHEMATICS

Z. I. KOZLOVA

ON SOME APPLICATIONS OF THE PRINCIPLE OF COMPARISON OF INDICES AND OF SEPARABILITY LAWS OF THE SPACE \(J^{\omega_\nu}\)

(Presented by Academician V. L. Kantorovich, 8 I 1969)

We shall consider functions defined at all points of the basic space \(\mathfrak R\) and taking values in some ordered set \(T=(\beta)\). The class of such functions \(\Lambda=(\beta(x))\) is called a class of transfinite functions, or a class of indices, if:
1) for whatever homeomorphic transformation \(y=\varphi(x):\mathfrak R\to\mathfrak R\), from \(\beta(x)\in\Lambda\) it follows that \(\beta(\varphi(x))=\beta'(x)\in\Lambda\);
2) from \(\beta(x)\in\Lambda\) and \(\beta'(x,y)=\beta(x)\) for every \(y\in\mathfrak R_y\) it follows that \(\beta'(x,y)\in\Lambda\).

Let \(\beta_1 * \beta_2\) be some order relation between elements of the set \(T\) (for example, \(\beta_1=\beta_2\), \(\beta_1\geqslant\beta_2\), \(\beta_1\ne\beta_2\)). We shall say that the class of transfinite indices \(\Lambda\) satisfies the \(\theta\)-principle of comparison of indices with defining relation \(*\), if, whatever \(\beta_1(x), \beta_2(x)\in\Lambda\) may be, the set \([\beta_1(x)*\beta_2(x)]\) of points \(x\) at which the relation \(\beta_1(x)*\beta_2(x)\) holds is a set of class \(\theta\).

Example 1. Let the space \(J^{\omega_\nu}\) be ordered lexicographically. Add to this space one more point \(\{\omega_\nu,0,0,\ldots,0,\ldots\}\), which we shall regard as following all points of the given space. Let \(E\in(B_\alpha)\subset\mathfrak R J_{xy}^{\omega_\nu}\). Put \(\beta(x,y)=y\), if \((x,y)\in E\); \(\beta(x,y)=\{\omega_\nu,0,0,\ldots,0,\ldots\}\), if \((x,y)\in CE\). Let \(\beta_1(x,y)\) be the index of the set \(E_1\in(B_\alpha)\), \(\beta_2(x,y)\) the index of the set \(E_2\in(B_\alpha)\), for any \(\alpha<\omega_{\nu+1}\); \([\beta_1(x,y)*\beta_2(x,y)]=[\beta_1(x,y)\leqslant\beta_2(x,y)]\). Then the set \(V=[\beta_1(x,y)*\beta_2(x,y)]\in(B_\alpha)\), i.e. the family of transfinite functions \(\Lambda=(\beta(x,y))\) satisfies the \(B_\alpha\)-principle of comparison of indices with defining relation \(*\).

Example 2. Let
\(\beta(x,y)=(\omega_\nu)T_{\mathfrak M}\operatorname{Ind}(x|\{\Delta_i\})\), where \(\mathfrak M=(M_i)\cup\bigcup(M_{(i_\alpha)\gamma})\) is an \((\omega_\nu)R_M\)-family of bases; \(\Phi_M\equiv\bigcup_i\); \((\Delta_i)_i\) is a regular family of Baer rectangles of the space \(J_{xy}^{\omega_\nu}\); \([\beta_1(x,y)*\beta_2(x,y)]=[\beta_1(x,y)<\beta_2(x,y)]\). Then the set \(V=[\beta_1(x,y)*\beta_2(x,y)]\in CT\mathfrak M(B)\), where \((B)\) is the class of \(B\)-sets of the space \(J_{xy}^{\omega_\nu}\), i.e. the class of transfinite functions \(\Lambda=(\beta(x,y))\) satisfies the \((\omega_\nu)CA\)-principle of comparison of indices with defining relation \(*\). Hence it follows that if
\[ [\beta_1(x,y)*\beta_2(x,y)]=[\beta_1(x,y)\ne\beta_2(x,y),\ \beta_2(x,y)<\omega_{\nu+1}], \]
i.e.
\[ V=[\beta_1(x,y)*\beta_2(x,y)]=([\beta_1(x,y)<\beta_2(x,y)]\cup[\beta_2(x,y)<\beta_1(x,y)])\cap[\beta_1(x,y)<\omega_{\nu+1}], \]
and also, if \(\beta_2(x,y)=\min\{\beta_1(x,y)+1,\omega_{\nu+1}\}\),
\[ [\beta_1(x,y)*\beta_2(x,y)]=[\beta_1(x,y)<\beta_2(x,y)], \]
i.e.
\[ V\equiv[\beta_1(x,y)*\beta_2(x,y)]=[\beta_1(x,y)<\beta_2(x,y)], \]
then the class of transfinite functions \(\Lambda=(\beta(x,y))\) satisfies the \((\omega_\nu)CA\)-principle of comparison of indices with the corresponding defining relation \(*\).

Let to a point \(x\in\mathfrak R\) there be assigned a set \(H_x\subset\mathfrak R\), and let
\[ U=[\beta_1(x)*\beta_2(x')\mid x'\in H_x], \]
i.e.
\[ x\in U\equiv(\forall x'\in H_x)[\beta(x)*\beta_2(x')]. \]
Let \(K(\Lambda)\) be the class of sets representable in the form \([\beta(x)=\omega_{\nu+1}]\). A. A. Lyapunov \((^1)\) proved the following proposition:

Lemma. Let \(\Lambda=(\beta(x))\) be a class of indices, regular \((^2,^3)\) relative to the class \(K(\Lambda)\), satisfying the \(CK(\Lambda)\)- or \(BK(\Lambda)\)-principle

comparison of indices with defining relation \(*\), where the class \(K(\Lambda)\) is such that, if \(M \in K(\Lambda)\), then \(\operatorname{pr}_{x} M \in K(\Lambda)\), and let to each point \(x\) of some set \(E \in BK(\Lambda)\) there be assigned a set \(H_x\) in such a way that \(\bigcup_x x \times H_x \in BK(\Lambda)\). Then, if \(\beta_1(x) \in \Lambda,\ \beta_2(x) \in \Lambda\), the set

\[ U=\bigl[\beta_1(x)*\beta_2(x') \mid x' \in H_x\bigr] \]

is a set of class \(CK(\Lambda)\).

We note a number of propositions following from this lemma.

1. The union of the Mazurkiewicz sets of all constituents of an \((\omega_\nu)CA\)-set of the space \(J_{xy}^{\omega_\nu}\) is also an \((\omega_\nu)CA\)-set.

2. The Mazurkiewicz set of any \(B_\alpha\)-set of the space \(J_{xy}^{\omega_\nu}\), for \(\alpha < \omega_{\nu+1}\), is a \(CA_\alpha\)-set.

3. The set of points of minimal index of every nonempty \((\omega_\nu)CA\)-set of the space \(J_{xy}^{\omega_\nu}\) is an \((\omega_\nu)CA\)-set.

4. The set of points of transfinite uniqueness of an \((\omega_\nu)CA\)-set of the space \(J_{xy}^{\omega_\nu}\) is an \((\omega_\nu)CA\)-set.

5. The set of points of unique index of an \((\omega_\nu)CA\)-set of the space \(J_{xy}^{\omega_\nu}\) is an \((\omega_\nu)CA\)-set.

6. There exists an \((\omega_\nu)CA\)-surface \(S \subset J_{xyz}^{\omega_\nu}\), uniform for all \((\omega_\nu)CA\)-curves.

Let \(E \subset \mathfrak{R}_{xy}\) be a set of some class \(K\), all of whose sections parallel to a given direction possess some structural property \(\bar S\). The question arises whether there exists such a \(BK\)-set \(\theta \supset E\), all of whose sections parallel to the same direction possess the structural property \(\bar S\). Problems of this kind have been called problems on the covering of plane sets. They were solved for the class of \(A\)-sets of the Baire space \(J\) by various authors \((^{4-9})\). Recently A. A. Lyapunov obtained, in the theory of operations on sets, two general theorems on the covering of plane sets. They are formulated for a countable space of indices, but carry over without change to an arbitrary space of indices. Let \(P_x=x\times J_y^{\omega_\nu}\).

First theorem on the covering of plane sets. Let \(H\) be an \(\mathfrak{A}_1\)-regular and \(\bar H\)-topological property of bases \((^{10})\), and let \(E \subset J_{xy}^{\omega_\nu}\) be an \((\omega_\nu)A\)-set. The set of all sets \(E\cap P_x\) possessing the property \(\bar H\), \(E_{\bar H}=E\). Then there exists a \(B\)-set \(S \supset E\) such that all sets \(S\cap P_x\) possess the property \(\bar H\).

Second theorem on the covering of plane sets. Let \(H\) be an \(\mathfrak{A}_1\)-regular and \(\bar H\)-topological property of bases; let \(E \subset J_{yx}^{\omega_\nu}\) be an \((\omega_\nu)A\)-set, and \(E_{\bar H} \subset E\) the union of all sets \(E\cap P_x\) possessing the property \(\bar H\). Then there exists a set \(S \supset E_{\bar H}\) of class \((\omega_\nu)CA_{\aleph_0}\) such that all sets \(S\cap P_x\) possess the property \(\bar H\).

We have proved the \(\mathfrak{A}_1\)-regularity of the properties \(H_p\) \((2\le p<\omega)\), \(H_{\aleph_\sigma}\), \(H_{t'}\) for \(\tau=\aleph_\nu\le \aleph_\nu=\tau'\), \(\tau\) a strongly inaccessible cardinal number \((^{10})\). We also note the \(\mathfrak{A}_1\)-regularity of the following properties: \(H_{\aleph_0}\) (to contain an uncountable number of different chains of the base \(\mathfrak{A}_1\) (for \(\tau=\aleph_0\)—the theorem of C. Mazurkiewicz and W. Sierpiński \((^{11})\), the author \((^{12})\)); \(H_{\mathrm{clr}\,\alpha}\) (to contain a family of chains of the given base that is not a dispersed set of bounded index \(\le \alpha\) (the author \((^{8})\) for \(\tau=\aleph_0\), Z. I. Evdokimova for \(\tau>\aleph_0\)); \(H_C(H_{|C|})\) (to contain a family of chains of the given base that is not \(\tau\)-compact (with \(\tau\)-compact closure) (the author \((^{8})\) for \(\tau=\aleph_0\), Z. I. Evdokimova for \(\tau>\aleph_0\)); \(H_{\mathrm{abs}\,\alpha}\) (to contain a family of chains of the given base that is not a dispersed family of \(\tau\)-compact sets of bounded index \(\le \alpha\) (the author \((^{9})\) for \(\tau=\aleph_0\), Z. I. Evdokimova for \(\tau>\aleph_0\)); \(H_{\mathrm{red}\,\alpha}\) (to contain a set of chains of the given base that is not a reducible set of index \(\le \alpha\) (Z. I. Evdokimova)).

The formulated theorems on the covering of plane \((\omega_\nu)A\)-sets are valid—

valid for the following topological properties \(\bar H\): \(\bar H_2\), \(\bar H_p\) \((2<p<\omega)\), \(\bar H\), \(\bar H_{[c]}\), \(\bar H_{\mathrm{red}\,\alpha}\).

For the space \(J_{xx}^{\omega_\nu}\) the theorems proved by the author \((^{8,9})\) for the space \(J\) also hold:

I. Every \((\omega_\nu)A\)-set \(\mathscr E \subset J_{xy}^{\omega_\nu}\) for which all the sets \(\mathscr E \cap P_x\) are scattered sets of index \(\leq \alpha < \omega_{\nu+1}\) can be covered by the same kind of \(B\)-set of this space.

II. Every \((\omega_\nu)A\)-set \(\mathscr E \subset J_{xy}^{\omega_\nu}\) for which each of the sets \(\mathscr E \cap P_x\) is the union of a scattered family of sets with \(\tau\)-compact closure of index \(\leq \alpha < \omega_{\nu+1}\) can be covered by a \(B\)-set \(H \subset J_{xy}^{\omega_\nu}\), for which all sets \(H \cap P_x\) are sets of absolute first class of subclass not exceeding \(\alpha\).

III. Every \((\omega_\nu)A\)-set \(\mathscr E \subset J_{xy}^{\omega_\nu}\) for which all the sets \(\mathscr E \cap P_x\) are sets of absolute first class of subclass \(\leq \alpha < \omega_{\nu+1}\) can be covered by the same kind of \(B\)-set of this space.

Let us note that the closure \(E^{(y)}\) of an \((\omega_\nu)A\)-set \(E \subset J_{xy}^{\omega_\nu}\) in the direction \(J_{xy}^{\omega_\nu}\) is an \((\omega_\nu)A\)-set.

A. A. Lyapunov, in the case of a countable space of indices, also proved general theorems on sections, projection, and representation of \(B\)-sets. They also hold for the space \(J_{xy}^{\omega_\nu}\).

Section theorem. Let \(\bar H\) be an \(\mathfrak A_1\)-regular and \(\bar H\)-topological property of bases; \(E \subset J_{xy}^{\omega_\nu}\) is a \(B\)-set. Then the set \(E_H \subset E\), which is the union of all sets \(E \cap P_x\) possessing the property \(\bar H\), is an \((\omega_\nu)CA\)-set.

Projection theorem. If \(\bar H\) is an \(\mathfrak A_1\)-regular and \(\bar H\)-topological property of bases, \(E \subset J_{xy}\) is a \(B\)-set, \(E_{\bar H} \subset E\) is the union of all sets \(E \cap P_x\) possessing the property \(\bar H\), then \(\operatorname{pr}_x E_{\bar H} \subset (\omega_\nu)CA\).

Representation theorem. If \(E \subset J_{xy}^{\omega_\nu}\) is such a \(B\)-set that the set of all points of the set \(E \cap P_x\) possessing the topological property \(\bar H\), \(E_{\bar H}=E\), and moreover the property of bases \(H\) is \(\mathfrak A_1\)-regular, then \(\operatorname{pr}_x E \in (B)\).

As the property \(\bar H\), one may take any of the topological properties listed above, and also the property \(\bar H_F\) (to be a closed set) in the first two theorems.

Volgograd Pedagogical Institute
named after A. S. Serafimovich

Received
30 XII 1968

CITED LITERATURE

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