UDC 513.831

B. A. PASYNKOV

ON BICOMPACTA WITH NONCOINCIDING DIMENSIONS

(Presented by Academician P. S. Aleksandrov, 18 XI 1969)

This note adjoins our joint note with I. K. Lifanov (¹) (see the beginning of (¹)).

1. We generalize the construction described in Sections 3 and 4 of note (¹).

(a) Suppose we are given: a bicompactum \(X\), a pair of its closed subsets \(F_1\) and \(F_2 \supset F_1\), bicompacta \(\Phi_1\) and \(\Phi_2\), a continuous mapping \(f\) of the bicompactum \(\Phi_1\) onto the bicompactum \(F_1\), and a continuous mapping \(g\) of the bicompactum \(\Phi_0\) onto the bicompactum \(\Phi_1\).

Consider the product \(X \times \Phi_1\). Denote by \(p\) the natural projection of \(X \times \Phi_1\) onto \(X\). The bicompactum \(\Phi_1\) may be identified with the graph \(\Psi \subset X \times \Phi_1\) of the mapping \(f: \Phi_1 \to F_1 \subset X\). Denote by \(q\) the mapping of the product \(X \times \Phi_0\) onto the product \(X \times \Phi_1\) which assigns to the point \((x,\varphi_0)\), \(x \in X\), \(\varphi_0 \in \Phi_2\), the point \((x,g(\varphi_0))\). Consider the bicompactum \(q^{-1}p^{-1}F_2 = F_2 \times \Phi_0 \subset X \times \Phi_0\) and its decomposition \(v\). The elements of \(v\) are: 1) the individual points of the set \(F_2 \times \Phi_0 \setminus q^{-1}\Psi\), and 2) the sets \(q^{-1}(x,\varphi_1)\) for \((x,\varphi_1) \in \Psi \equiv \Phi_1\). We denote the decomposition space \(v\) by \(E = E(X,F_2,F_1,\Phi_1,\Phi_0,f,g)\). Denote by \(v\) the natural mapping of the bicompactum \(F_2 \times \Phi_0\) onto \(E\). Denote by \(\mu\) the mapping of the bicompactum \(E\) into \(X \times \Phi_1\) satisfying the condition \(\mu \cdot v = q\). Obviously, on the set \(\mu^{-1}\Psi\) the mapping \(\mu\) is a homeomorphism, and therefore the set \(\mu^{-1}\Psi\) may be identified with the bicompactum \(\Phi_1 \equiv \Psi\). The mapping \(\mathfrak{P} = p \cdot \mu\) will be called the projection of the bicompactum \(E\) onto the bicompactum \(X\).

(b) Let a bicompactum \(X\) and a system of pairs \((F_2^\theta,F_1^\theta)\) of its closed subsets \(F_1^\theta\) and \(F_2^\theta \supset F_1^\theta\), \(\theta \in \Theta\), be given. Let, in addition, a bicompactum \(\chi\) be given, and in it a disjoint system of open sets \(C_\theta\), \(\theta \in \Theta\), each of which decomposes into the disjoint sum of open-closed bicompacta \(\Phi_0^\alpha\), \(\alpha \in \mathfrak{A}_\theta\). Suppose that for each \(\alpha\) a mapping \(g_\alpha\) of the bicompactum \(\Phi_0^\alpha\) onto some bicompactum \(\Phi_1^\alpha\), and a mapping \(f_\alpha\) of the bicompactum \(\Phi_1^\alpha\) onto the bicompactum \(F_1^\theta\), \(\alpha \in \mathfrak{A}_\theta\), \(\theta \in \Theta\), are defined. Take the decomposition \(v\) of the bicompactum \(Y = X \times \chi \setminus \bigcup_\theta (X \setminus F_2^\theta \times C_\theta)\) into the individual points of the set \(X \times (\chi \setminus \bigcup_\theta C_\theta)\) and the inverse images of the points of the bicompactum \(E_\alpha = E(X,F_2^\theta,F_1^\theta,\Phi_1^\alpha,\Phi_0^\alpha,f_\alpha,g_\alpha)\) under the mapping \(v_\alpha: F_2^\theta \times \Phi_0^\alpha \to E_\alpha\) (see (a)) for all \(\alpha \in \mathfrak{A}_\theta\) and \(\theta \in \Theta\). We denote the decomposition space \(v\) by

\[ E = E(X,\chi,\{F_2^\theta,F_1^\theta,\Phi_1^\alpha,\Phi_0^\alpha,f_\alpha,g_\alpha;\ \alpha \in \mathfrak{A}_\theta\},\ \theta \in \Theta). \tag{*} \]

Denote by \(v\) the natural mapping of the bicompactum \(Y\) onto the bicompactum \(E\). Denote by \(p\) the natural projection of \(X \times \chi\) onto \(X\). The mapping \(\mathfrak{P}: E \to X\), satisfying the relation \(\mathfrak{P}\cdot v = p\), will be called the projection of \(E\) onto \(X\). If the bicompacta \(v(F_2^\theta \times \Phi_0^\alpha)\) are identified with the bicompacta \(E_\alpha\), then on \(E_\alpha\) the projection \(v\) coincides with the projection \(v_\alpha: E_\alpha \to X\).

Remark 1. a) If the bicompactum \(Y\) is the image under a mapping \(f\) of the bicompactum \(X\), then below a point \(y \in Y\) will often be denoted by one (or several) points of its inverse image \(f^{-1}y\). b) Let the bicompactum \(Y = \{y\}\) be the image under a finite-to-one mapping \(f\) of the bicompactum \(X\), which, in turn, is the product \(X_1 \times X_2\) of ordered bicompacta \(X_1 = \{x_1\}\) and \(X_2 = \{x_2\}\). Then by \(+^{1}/_{2}(y_0,Y,j)\), respectively \(-^{1}/_{2}(y_0,j) = -^{1}/_{2}(y_0,Y,j)\), \(j=1,2\), we denote the set
\[ \{y:\ y \in f\{(x_1,x_2);\ x_j \ge \max x_j^0,\ \text{respectively } \le \min x_j^0,\ \text{for } (x_1^0,x_2^0) \in f^{-1}y_0\}\}. \]

2. Bicompacta \(T_1\) and \(S_1\). Consider ordered bicompacta

\(L_k=\{l_k\}\) with such points \(a_k\in L_k\) that the first axiom of countability holds at \(a_k\), \(k=0,1\). Consider the products \(L_k\times C_k\), where \(C_k=\{c_k\}\) denotes the Cantor perfect set, \(k=0,1\). By \(\omega_k\) we denote the decomposition of \(L_k\times C_k\) whose elements are pairs of endpoints of intervals adjacent to the Cantor set \(a_k\times C_k\), and the individual points not belonging to these pairs. The decomposition space \(\omega_k\) will be denoted by \(\Gamma_k'\), and the natural mapping of \(L_k\times C_k\) onto \(\Gamma_k'\) by \(\omega_k\) (as also the corresponding decomposition). By \(M_k'\) we denote the set of those points of the bicompactum \(\Gamma_k'\) whose inverse image under the mapping \(\omega_k\) consists of two points. The set \(Q_k^1=\omega_k(a_k\times C_k)\) is a segment, on which the countable set \(M_k'\) is everywhere dense. There exists a homeomorphism \(h\) of the segment \(Q_0^1\) onto the segment \(Q_1^1\), for which \(hM_0'=Q_1^1\setminus M_1'\). The decomposition space of the discrete sum of the bicompacta \(\Gamma_0'\) and \(\Gamma_1'\) into the individual points of the set \((\Gamma_0'\cup\Gamma_1')\setminus(Q_0^1\cup Q_1^1)\) and the pairs of points \((x,h(x))\), \(x\in Q_0^1\), will be denoted by \(\Gamma=\Gamma(L_0,L_1,a_0,a_1)\). The natural mapping of \(\Gamma_0'\cup\Gamma_1'\) onto \(\Gamma\) will be denoted by \(\omega\). Let \(Q_\Gamma^1=\omega(Q_0^1\cup Q_1^1)\), \(\Gamma_k=\omega\Gamma_k'\), \(M_k=\omega M_k'\), \(k=0,1\), \(N=Q_\Gamma^1\setminus(M_0\cup M_1)\).

Mark the following pairs \((x,F)\) of points \(x\in\Gamma\) and closed subsets \(F\supset x\) in \(\Gamma\): a) \(x\in N\), \(F=\Gamma\); b) \(x\in M_0\cup M_1\), \(F=+^{1/2}(x,\Gamma_0,2)\cup+^{1/2}(x,\Gamma_1,2)\); \(x\in M_0\cup M_1\), \(F=-^{1/2}(x,\Gamma_0,2)\cup-^{1/2}(x,\Gamma_1,2)\); c) \(x=(a_k,c_k)\in M_{\bar k}*\), the set \(F\) satisfies the following conditions: \(F\) is the closure of an open subset of \(\Gamma\), \(F\cap\Gamma_{\bar k}=x\), and for some points \(l_k'<a_k\) and \(l_k''>a_k\) the set \(\{(l_k,c_k):l_k'\le l_k\le l_k''\}\) is contained in the interior of \(F_k\), \(k=0,1\). The set of marked pairs will be denoted by \(B'\).

Consider the bicompacta \(\chi_1\) and \(\chi_2\) from (1) and establish a one-to-one correspondence: 1) between the set \(B'\) and the set \(\Theta\) of item 5 of remark (1); 2) between the set \(B'\) and the set \(\Theta'\) of item 51 of remark (1). Fix some mapping \(g\) of the Cantor perfect set \(C\) onto the segment \(Q^1\). Put the bicompactum \(T_1=T_1(L_0,L_1,a_0,a_1)\), respectively \(S_1=S_1(L_0,L_1,a_0,a_1)\), equal to the bicompactum \(E\) from formula \((*)\), where \(X=\Gamma\); \(\chi=\chi_1\), respectively \(\chi=\chi_2\); \(F_2^\theta=F^\theta\), \(F_1^\theta=x_\theta\), where \((x_\theta,F^\theta)\) is the element of the set \(B'\) corresponding to the element \(\theta\) of the set \(\Theta\), respectively \(\Theta'\); \(\Phi_1^\alpha=Q^1\); \(\Phi_0^\alpha=C_\alpha\); \(g_\alpha=g\); \(f_\alpha\) is a mapping of \(\Phi_1^\alpha\) to the point \(x_\theta\). The projections of the bicompacta \(T_1\) and \(S_1\) onto \(\Gamma\) (see item 1) will be denoted respectively by \(\widetilde\omega_T\) and \(\widetilde\omega_S\).

Proposition 1. If the set of points \(x\in L_k\) for which \(\operatorname{ind}_x L_k=1\) is everywhere dense in \(L_k\), \(k=0,1\), then \(\dim T_1=\dim S_1=1\), \(\operatorname{ind} T_1=\operatorname{ind} S_1=2\). The set \(\widetilde\omega_S^{-1}Q_\Gamma^1=\widetilde\omega_S^{-1}T_0\cap\widetilde\omega_S^{-1}T_1\) has type \(G_\delta\) in \(S_1\), and \(\operatorname{ind}\widetilde\omega_S^{-1}\Gamma_k=1\), \(k=0,1\). If the bicompacta \(L_0\) and \(L_1\) possess the first axiom of countability, then the bicompactum \(S_1\) will also possess the first axiom of countability.

Corollary. The bicompactum \(S_1=S_1(Q^1,Q^1,1,1)\) with the first axiom of countability can be represented as the sum of two subbicompacta \(S'\) and \(S''\) of dimension \(\operatorname{ind} S'=\operatorname{ind} S''=1\), whose intersection has type \(G_\delta\) in \(S_1\), but \(\operatorname{ind} S_1=2\).

Remark 2. If the bicompacta \(L_0\) and \(L_1\) possess the first axiom of countability, then the bicompactum \(S_1(L_0,L_1,a_0,a_1)\) is an irreducible image of a zero-dimensional bicompactum with the first axiom of countability under some mapping \(\lambda=\lambda(L_0,L_1,a_0,a_1)\).

3. Bicompacta \(T^2\) and \(S^2\). By \(R=\{r\}\) and \(I=\{i\}\) we denote respectively the sets of rational and irrational points of the segment \(Q^1=\{t,0\le t\le1\}\). Represent the set \(R\) as the disjoint sum of two sets \(R_0=\{r_0\}\) and \(R_1=\{r_1\}\) everywhere dense in \(Q^1\). Consider the square \(Q^2=Q^1\times Q^1\). We shall call marked the following pairs \((x,F)\) of points \(x\in Q^2\) and closed subsets \(F\supset x\) in \(Q^2\)**: a) \(x=(i',i'')\), \(F=Q^2\); b) \(x=(k,t)\), \(t\in R_k\cup I\), \(F=Q^2\), \(k=0,1\); c) \(x=(t^1,t^2)\), \(0<t^1<1\), \(t^j\in\)

* Everywhere in the note \(k\) is equal either to \(0\) or to \(1\), and if \(k=0\), then \(\bar k=1\), while if \(k=1\), then \(\bar k=0\).

** If the pair 1) \((x,+^{1/2}(x,j))\) or 2) \((x,+^{1/2}(x,1)\cap+^{1/2}(x,2))\) is marked, then in each of the cases 1) and 2) the pairs obtained from the indicated pair by replacing, in its second element, some (or all) signs \(+\) by the sign \(-\), are also considered marked.

\(\in R,\ t_k \in I,\ j=1,2,\ j'=2,1,\ F=+^{1/2}(x,j)\); d) \(x=(k,r_k),\ F=+^{1/2}(x,2)\), \(k=0,1\); e) \(x=(r^1,r^2),\ 0<r^1<1,\ F=+^{1/2}(x,1)\cap +^{1/2}(x,2)\).

Denote the set of all marked pairs by \(B_1\). Denote by \(B_2\) the set of points \(x\) of the form \((r^1,r^2)\), \(0<r^1<1,\ r^j\in R_0,\ r^{j'}\in R_1,\ j=1,2,\ j'=2,1\). Denote the sum \(B_1\cup B_2\) by \(B\).

We construct the bicompactum \(T^2\). Represent the set of all ordinal numbers \(\alpha<\omega(\mathfrak c)\) as the disjoint sum of sets \(\mathfrak A_\theta\) of cardinality \(\mathfrak c\), \(\theta\in\Theta\), where the cardinality of \(\Theta\) is also equal to \(\mathfrak c\). Establish a one-to-one correspondence between the sets \(B\) and \(\Theta\). Here the subset of \(\Theta\) corresponding to the set \(B_j\) will be denoted by \(\Theta_j,\ j=1,2\). Let
\[ \mathfrak A_j=\bigcup_{\theta\in\Theta_j}\mathfrak A_\theta, \]
\(j=1,2\). To each \(\alpha\in\mathfrak A_\theta,\ \theta\in\Theta_1\), assign a Cantor perfect set \(C_\alpha\). To each \(\alpha\in\mathfrak A_\theta,\ \theta\in\Theta_2\), there corresponds a point \(x_\theta=(r^1,r^2)\in B_2\). Put \(a_0=a_1=x_\theta\) and \(L_0^\theta=\{(r^1,t):0\le t\le1\},\ L_1^\theta=\{(t,r^2):0\le t\le1\}\). Denote by \(T_1^\alpha\) the bicompactum \(T_1(L_0^\theta,L_1^\theta,x_\theta,x_\theta)\), by \(\widetilde{\omega}_{T\alpha}\) the projection of \(T_1^\alpha\) onto \(L_0^\theta\cup L_1^\theta\), and by \(\lambda_{T\alpha}\) an irreducible mapping of some zero-dimensional bicompactum \(T_0^\alpha\) onto \(T_1^\alpha\). Put
\[ \chi_3=\bigcup_{\alpha\le\omega(\mathfrak c)}\alpha\ \cup\ \bigcup_{\alpha\in\mathfrak A_\theta,\ \theta\in\Theta_1} C_\alpha\ \cup\ \bigcup_{\alpha\in\mathfrak A_2} T_0^\alpha . \]
The open sets in \(\chi_3\) are taken to be: the open subsets of the bicompacta \(C_\alpha,\ \alpha\in\mathfrak A_1\), and \(T_0^\alpha,\ \alpha\in\mathfrak A_2\); isolated numbers \(\alpha\); sets of the form
\[ \bigcup_{\alpha'<\alpha\le\alpha''}\alpha\ \cup\ \bigcup_{\alpha'<\alpha<\alpha''}\bigl(C_\alpha\cup T_0^\alpha\bigr) \]
for limit numbers \(\alpha''\).

The bicompactum \(\chi_3\), obviously, is zero-dimensional. Finally, set \(T^2\) equal to the bicompactum \(E\) from formula \((*)\), where \(X=Q^2;\ \chi=\chi_3\ \otimes\), and where: 1) for \(\theta\in\Theta_1\) the set \(F_2^\theta\) denotes the closed set \(F=F_\theta\), while the set \(F_1^\theta\) is the point \(x_\theta\) from the marked pair (i.e., from the element of the set \(B_1\)) corresponding to the index \(\theta\); the bicompactum \(\Phi_1^\alpha\) is the square \(Q^2;\ \Phi_0^\alpha=C_\alpha;\ g_\alpha\) coincides with some fixed mapping \(g\) of the Cantor perfect set \(C\) onto the square \(Q^2\); \(f_\alpha\) denotes the mapping of \(Q^2\) to the point \(x_\theta\); 2) for \(\theta\in\Theta_2\) we take \(F_2^\theta=Q^2,\ F_1^\theta=L_0^\theta\cup L_1^\theta,\ \otimes\otimes\ \Phi_1^\alpha=T_1^\alpha,\ \Phi_0^\alpha=T_0^\alpha,\ f_\alpha=\widetilde{\omega}_{T\alpha},\ g_\alpha=\lambda_{T\alpha}\).

We construct the bicompactum \(S^2\). The product \(P=Q_u^1\times Q_\theta^1\times D\) of the interval \(Q_u^1=\{u:0\le u\le1\}\), the interval \(Q_\theta^1=\{\theta:0\le\theta\le1\}\), and the pair of points \(D=\{0,1\}\) will be considered in the natural lexicographic ordering. Establish a one-to-one correspondence between the set \(Q_0^1\) and the set \(B\). Denote by \(\Theta_j\) the subset of the set \(Q_\theta^1\) corresponding to the set \(B_j,\ j=1,2\). Let further
\[ \mathfrak A_\theta=\{\alpha=(u,\theta):0\le u\le1\},\quad \theta\in Q_\theta^1;\qquad \mathfrak A_j=\bigcup_{\theta\in\Theta_j}\mathfrak A_{\theta,j}=1,2. \]
To each \(\alpha\in\mathfrak A_1\) assign a Cantor perfect set \(C_\alpha\). Fix some mapping \(g:C\to Q^2\).

To each \(\alpha\in\mathfrak A_2\) there corresponds a point \(x_\theta=(r^1,r^2)\in B_2\). Denote by \(S_1^\alpha\) the bicompactum \(S_1(L_0^\theta,L_1^\theta,x_\theta,x_\theta)\) (see the construction of \(T^2\)), by \(\widetilde{\omega}_{S\alpha}\) the projection of \(S_1^\alpha\) onto \(L_0^\theta\cup L_1^\theta\), and by \(\lambda_{S\alpha}\) an irreducible mapping of some zero-dimensional bicompactum with the 1st axiom of countability \(S_0^\alpha\) onto \(S_1^\alpha\) \(\boxtimes\). Put
\[ \chi_4=P\cup\bigcup_{\alpha\in\mathfrak A_1} C_\alpha\cup\bigcup_{\alpha\in\mathfrak A_2} S_0^\alpha . \]
The open sets in \(\chi_4\) are taken to be: the open subsets of the bicompacta \(C_\alpha,\ \alpha\in\mathfrak A_1\), and \(S_0^\alpha,\ \alpha\in\mathfrak A_2\); the points \((0,0,0)\) and \((1,1,1)\), sets of the form
\[ \bigl((u_1,\theta_1,0),(u_2,\theta_2,1)\bigr)\cup \bigcup C_\alpha\cup \bigcup S_0^\alpha, \]
where the first term is an interval, and the summation in the second and third terms is taken over those \(\alpha=(u,\theta)\) for which
\[ (u_1,\theta_1,0)<(u,\theta,0)<(u,\theta,1)<(u_2,\theta_2,1). \]
Set \(S^2\) equal to the bicompactum \(E\) from formula \((*)\), where \(X=Q^2;\ \chi=\chi_4\) (further one must again read the part of the construction of the bicompactum \(T^2\) from the sign \(\otimes\) to the sign \(\otimes\otimes\)). \(\Phi_1^\alpha=S_1^\alpha,\ \Phi_0^\alpha=S_0^\alpha,\ f_\alpha=\widetilde{\omega}_{S\alpha},\ g_\alpha=\lambda_{S\alpha}\).

Theorem 1. The bicompactum \(S^2\) has the 1st axiom of countability and
\[ \dim S^2=\dim T^2=\operatorname{ind} S^2=\operatorname{ind} T^2=2<\operatorname{Ind} S^2=\operatorname{Ind} T^2=3. \]

4. Bicompacta of PH. Denote by \(\zeta\) the mapping of the Cantor perfect set \(C=\{c\}\) onto the interval \(Q^1=\{v:0\le v\le1\}\), identifying

corresponding to pairs of endpoints of intervals adjacent to \(C\). The point \(c\) from \(C\) will be denoted, if necessary, by the corresponding point \(\xi c=v\) from \(Q^1\); moreover, if the point \(v\) corresponds to a pair of endpoints of some interval adjacent to \(C\), then (when this is needed) the left endpoint of this interval will be denoted by \(v_l\), and the right endpoint by \(v_r\).

By \(\chi_5\) we denote the lexicographically ordered (with the interval topology) product of a countable collection of Cantor perfect sets \(C_j=\{c\}\), \(j=1,2,\ldots\). Choose on \(Q^1\) a continuum of pairwise disjoint everywhere dense sets \(Q_\alpha\), \(\alpha\in\mathfrak A\). For a given bicompactum \(X=\{t\}\) of cardinality \(c\) we construct the bicompactum \(\Pi X\) as follows. Let \(h:X\to\mathfrak A\) denote a one-to-one correspondence between \(X\) and \(\mathfrak A\). The elements of the partition \(\omega\) of the product \(\chi_5\times X\) will be taken to be: (a) pairs of points
\[ ((c^1,\ldots,c^{n-1},v_l^n,1,1,1,\ldots),t) \]
and
\[ ((c^1,\ldots,c^{n-1},v_r^n,0,0,0,\ldots),t), \]
if there exists an index \(k\), \(1\le k<n\), such that \(\xi c^k\in Q_{h(t)}\), \(t\in X\); (b) individual points not entering, for any \(t\), into the pairs indicated in item (a). We denote the quotient space \(\omega\) by \(\Pi X\).

If the natural projection of \(\chi_5\times X\) onto \(X\) is denoted by \(p\), and the natural mapping of \(\chi_5\times X\) onto \(\Pi X\) by \(\omega\), then there exists a mapping \(\pi:\Pi X\to X\) satisfying the condition \(p=\pi\cdot\omega\). We shall call the mapping \(\pi\) the projection of \(\Pi X\) onto \(X\).

Theorem 2. If \(X\) is a compactum and \(\operatorname{ind} X=n\), then \(\dim \Pi X=n\), \(\operatorname{ind}\Pi X=n+1\), \(n=1,2,\ldots\). If, in addition, the compactum \(X\) is connected, then in \(\Pi X\) there are two such nowhere dense disjoint closed sets \(F_1\) and \(F_2\) that, for any open set \(O\subseteq[O]\subseteq X\setminus F_1\) or \(\subseteq X\setminus F_2\), one always has \(\operatorname{ind}\operatorname{fr}O\ge n-1\). The bicompactum \(\Pi X\) satisfies the first axiom of countability.*

5. The bicompactum \(T_1^2\). Let \(\Pi=\Pi Q^1\), where \(Q^1=\{t:0\le t\le 1\}\). By \(R=\{r\}\) denote the set of points of \(\chi_5\) not having the form \((c^1,\ldots,c^n,0,0,0,\ldots)\) or \((c^1,\ldots,c^n,1,1,1,\ldots)\). Obviously, \(R\) is everywhere dense in \(\chi_5\). Represent \(R\) as the disjoint sum of the sets \(R_0=\{r_0\}\) and \(R_1=\{r_1\}\) dense in \(\chi_5\). By \(0,1\), and \(I=\{i\}\) denote respectively the points \((0,0,0,\ldots)\) and \((1,1,1,\ldots)\) of \(\chi_5\), and the set \(\chi_5\setminus R\).

We note that on the segment \(Q^1\) there also are points \(0,1\) and the sets \(R\), \(R_0\), \(R_1\), and \(I\) (see item 3). Mark pairs \((x,F)\)—points \(x\in\Pi\) and closed sets \(F\ni x\) in \(\Pi\)—the same as in subitems a)–e) of item 3, with \(Q^2\) replaced by \(\Pi\) in subitems a) and b) (see Remark 1). The set of all marked pairs will be denoted by \(B_1\). The set of points \(x\in\Pi\) of the form \((r^1,r^2)\), \(0<r^1<1\), \(r^j\in R_0\), \(r^{j'}\in R_1\), \(j=1,2;\ j'=2,1\), will be denoted by \(B_2\). The sum \(B_1\cup B_2\) will be denoted by \(B\).

The further construction of \(T_1^2\) is analogous to the construction of \(T_2\), but instead of the bicompactum \(\chi_3\) one uses the “longer” bicompactum \(\chi_6\), constructed in the same way as \(\chi_3\), but with the use of ordinals \(\alpha\le \omega(2^c)\).**

Theorem 3. The inequalities
\[ \dim T_1^2=1<\operatorname{ind}T_1^2=2<\operatorname{Ind}T_1^2 \]
hold.

Faculty of Mechanics and Mathematics
M. V. Lomonosov Moscow State University

Received
13 XI 1969

REFERENCES

1. B. A. Pasynkov, I. K. Lifanov, DAN, 192, No. 2 (1970).
2. V. V. Filippov, DAN, 192, No. 2 (1970).
3. I. K. Lifanov, V. V. Filippov, DAN, 192, No. 1 (1970).

* The existence of bicompacta satisfying the first axiom of countability with \(\dim X=1\), \(\operatorname{ind}X>1\) was established earlier, by another method, by V. Filippov (2).

** Initially, in the construction of \(T_1^2\), instead of \(\Pi\) a bicompactum from (3) was used, to which I. K. Lifanov drew my attention.