UDC 513-83

MATHEMATICS

I. K. LIFANOV

ON THE DIMENSION OF THE PRODUCT OF ORDERED CONTINUA

(Presented by Academician P. S. Aleksandrov on 26 XI 1966)

Let \(X_1, X_2, \ldots, X_n\) be ordered continua*, and let
\[ P_n=\prod_{i=1}^{n} X_i \]
be their topological product. The main result of this note is the following

Theorem. For bicompacta \(P_n\) one necessarily has
\[ \dim P_n=\operatorname{ind} P_n=\operatorname{Ind} P_n=n. \]

For the proof of this proposition we shall need the following well-known result (see (2)).

Proposition. A normal space \(X\) has \(\dim X \ge n\) if there exist \(n\) pairs of closed sets \((C_1^i, C_2^i)\), \(C_1^i \cap C_2^i=\varnothing\) \((i=1,\ldots,n)\), such that for any partitions** \(B^i\) \((i=1,2,\ldots,n)\) one necessarily has
\[ \bigcap_{i=1}^{n} B^i \ne \varnothing . \]

If \(X\) is an ordered continuum, then by \(x_1\) and \(x_2\) we denote respectively the minimal and maximal points. We divide the proof of our theorem into several assertions.

Assertion 1. \(\dim P_n \ge n\).

We shall prove this assertion by induction on the number of factors. Let \(n=2\), i.e. \(P_2=X\times Y\), where \(X\) and \(Y\) are ordered continua. Then for any open cover \(\omega\) of the bicompactum \(P_2\) there exists a monotone \(\omega\)-mapping onto the square \(I^2\) \((I=[0,1])\), under which the sets \(X\times y_1\), \(X\times y_2\) and \(Y\times x_1\), \(Y\times x_2\) are mapped respectively onto opposite sides of the square. Put \(C_1^1=X\times y_1\), \(C_2^1=X\times y_2\), \(C_1^2=Y\times x_1\), \(C_2^2=Y\times x_2\). Let \(B^1\) be a partition for the pair \((C_1^1,C_2^1)\), and \(B^2\) a partition for the pair \((C_1^2,C_2^2)\). We shall show that
\[ B^1\cap B^2\ne \varnothing . \]
Thus we shall prove that \(\dim P_2 \ge 2\).

Suppose the contrary; let \(B^1\cap B^2=\varnothing\). Then take open sets \(G_1^1 \supset B^1\) and \(G_1^2 \supset B^2\) such that: 1) \([G_1^1]\cap [G_1^2]=\varnothing\), 2) \([G_1^i]\cap (C_1^i\cup C_2^i)=\varnothing\) \((i=1,2)\). Next consider open sets \(G_2^i\) such that \(B^i\subset G_2^i\subset [G_2^i]\subset G_1^i\). By the choice of the sets \(B^i\) we have \(X\setminus B^i=V_1^i\cup V_2^i\), where \(V_1^i,V_2^i\) \((V_1^i\cap V_2^i)=\varnothing\) are open and \(C_1^i\subset V_1^i\), while \(C_2^i\subset V_2^i\) \((i=1,2)\).

Consider the sets
\[ U_1=\Gamma\cap (V_1^1\cap V_1^2),\quad U_2=\Gamma\cap (V_1^1\cap V_2^2), \]
\[ U_3=\Gamma\cap (V_2^1\cap V_1^2),\quad U_4=\Gamma\cap (V_2^1\cap V_2^2), \]
where \(\Gamma=X\setminus [G_2^1]\setminus [G_2^2]\). Let us now consider the cover \(\omega\) of the bicompactum \(P_2\), consisting of the sets
\[ \{G_1^1, G_1^2, U_1, U_2, U_3, U_4\}, \]
and let \(f\) be a monotone \(\omega\)-mapping (for this cover) of \(P_2=X\times Y\) onto the square \(I^2\), under which \(X\times y_1\), \(X\times y_2\) and \(Y\times x_1\), \(Y\times x_2\) go respectively into opposite sides of the square \(I^2\). Then: 1) \(f(B^i)\cap f(C_j^i)=\varnothing\) \((i,j=1,2)\); 2) \(f(C_1^i)\cap f(C_2^i)=\varnothing\) \((i=1,2)\); 3) \(f(B^i)\) is a partition for the pair \(f(C_1^i), f(C_2^i)\). Let \(y\in f(B^1)\cap f(B^2)\). Then \(f^{-1}(y)\cap B^1\ne \varnothing\), \(f^{-1}(y)\cap B^2\ne \varnothing\), and since \(f\) is an \(\omega\)-mapping, \(f^{-1}(y)\) is necessarily contained both in \(G_1^1\) and in \(G_1^2\), while \(G_1^1\cap G_1^2=\varnothing\), a contradiction.

* A continuum is a connected bicompactum.

** A set \(B\) is called a partition for the pair \((C_1,C_2)\), \(C_1\cap C_2=\varnothing\), if \(X\setminus B=V^1\cup V^2\), \(V^1,V^2\) are open and \(V^1\cap V^2=\varnothing\), \(C_1\subset V^1\), \(C_2\subset V^2\).

We have obtained a contradiction, which proves that \(B^1 \cap B^2 \ne \phi\). Thus, assertion 1 for \(P_2\) is proved.

Let us outline the proof for the bicompactum \(P_n\) (i.e., for a bicompactum which is the product of \(n\) ordered continua). Let

\[ P_n=\prod_{i=1}^{n} X_i . \]

Denote the end points of the ordered continuum \(X_i\) respectively by \(x^{i}_{1}\) and \(x^{i}_{2}\). Consider the sets

\[ C^{i}_{1}=\prod_{\substack{k=1\\ k\ne i}}^{n} X_k \times x^{i}_{1}, \qquad C^{i}_{2}=\prod_{\substack{k=1\\ k\ne i}}^{n} X_k \times x^{i}_{2}. \]

The sets \(C^{i}_{1}\) and \(C^{i}_{2}\) \((i=1,2,\ldots,n)\) are closed and \(C^{i}_{1}\cap C^{i}_{2}=\phi\). It turns out that if the set \(B^i\) is a partition for the pair \((C^{i}_{1}, C^{i}_{2})\), then

\[ \bigcap_{i=1}^{n} B^i \ne \phi . \]

Suppose the contrary, namely let

\[ \bigcap_{i=1}^{n} B^i=\phi . \]

Then we construct such an open cover \(\omega\) of the bicompactum \(P_n\) that no element of this cover intersects all the sets \(B^i\) \((i=1,2,\ldots,n)\), and no element intersects both \(B^i\) and \(C^{i}_{1}\) or \(C^{i}_{2}\) \((i=1,2,\ldots,n)\). For the given cover \(\omega\) we construct such a monotone \(\omega\)-mapping \(f\) of the bicompactum \(P_n\) onto the \(n\)-dimensional cube \(I^n\) that the sets \(f(C^{i}_{1})\) and \(f(C^{i}_{2})\) are \((n-1)\)-dimensional opposite faces in \(I^n\). Since the mapping \(f\) is monotone and an \(\omega\)-mapping, the set \(f(B^i)\) is a partition for the corresponding pair of opposite \((n-1)\)-dimensional faces of the \(n\)-dimensional cube \(I^n\). Consequently,

\[ \bigcap_{i=1}^{n} f(B^i)\ne \phi, \]

which contradicts the fact that \(f\) is an \(\omega\)-mapping. Thus,

\[ \bigcap_{i=1}^{n} B^i\ne \phi, \]

and assertion 1 is proved.

Assertion 2. \(\operatorname{Ind} P_n \le n\).

Lemma 1. Let \(X\) be an ordered continuum, and let \(U\) be an arbitrary open subset of it; then

\[ \operatorname{ind}\operatorname{Fr} U=\operatorname{Ind}\operatorname{Fr} U=0. \]

Lemma 2. Let

\[ P=\bigcup_{i=1}^{s} X_i, \]

where \(X_i\) is an ordered continuum. Then

\[ \operatorname{ind} P=\operatorname{Ind} P=1. \]

Let

\[ P^{i}_{n}=\prod_{k=1}^{n} X^{i}_{k}, \]

where \(X^{i}_{k}\) is an ordered continuum. By \(x^{i}_{j_k}\) \((k=1,2,\ldots,n;\ j_k=1\ \text{or}\ 2)\) we denote the minimal and maximal points.

Definition 1. An \(l\)-dimensional face \((l<n)\) of \(P^{i}_{n}\) will be called

\[ F^{i}_{l(j_k)} = \prod_{m=1}^{l} X^{i}_{k_m} \times x^{i}_{j_1} \times \cdots \times x^{i}_{j_{k_1-1}} \times x^{i}_{j_{k_1+1}} \times \cdots \times x^{i}_{j_n}. \]

Definition 2. We shall say that \(P^{i_1}_{n}\) intersects \(P^{i_2}_{n}\) regularly with respect to \(X^{i_1}_{k}\) if

\[ X^{i_1}_{k}\times x^{i_1}_{j_1}\times \cdots \times x^{i_1}_{j_{k-1}}\times x^{i_1}_{j_{k+1}}\times \cdots \times x^{i_1}_{j_n} = X^{i_1}_{k}(j^{i_1}_{i}) \]

\((j_i=1\ \text{or}\ 2,\ i=1,2,\ldots,n;\ i\ne k)\) is equal to an \(X^{i_2}_{l}\)-factor for \(P^{i_2}_{n}\).

Definition 3. A collection \(\{P^{i}_{n}\}\) \((i=1,2,\ldots,s)\) will be called regular with respect to \(X^{1}_{k}\) if, for any \(P^{k}_{n}\) \((1\le k\le s)\), one can find such a sequence of bicompacta \(\{P^{i_l}_{n}\}\) \((i_l=1,\ldots,m;\ m<k)\) that \(P^{i_1}_{n}=P^{1}_{n}\), \(P^{i_1}_{n}\) intersects regularly with \(P^{i_2}_{n}\) with respect to \(X^{1}_{k}\), \(P^{i_2}_{n}\) intersects regularly with \(P^{i_3}_{n}\) with respect to \(X^{1}_{k}(j^{i_1}_{1})\), and so on; \(P^{i_{m-1}}_{n}\) with \(P^{i_m}_{n}=P^{k}_{n}\) with respect to

\[ X^{1}_{k}(j^{i_1}_{1})(j^{i_2}_{1})\cdots(j^{i_{m-2}}_{1}). \]

Definition 4. We shall say that \(P_n^{i_1}\) intersects \(P_n^{i_2}\) along some \(l\)-dimensional face \(F_{l(j_k)}^{i_1}\), if \(P_n^{i_1}\) intersects properly with \(P_n^{i_2}\) relative to \(X_{k_1}^{i_1}, \ldots, X_{k_l}^{i_1}\).

Lemma 3. Suppose we have such a sum of bicompacts \(\{P_n^i\}\) \((i=1,2,\ldots,s)\) that each \(P_n^{i_1}\) intersects with \(P_n^{i_2}\) along an \(l\)-dimensional face \((l<n)\). Then
\[ \operatorname{Ind}\left(\bigcup_{i=1}^{s} P_n^i\right)\leq n . \]

Proof. We shall carry out the proof by induction on the number \(n\). For a sum of bicompacts \(\{P_1^i\}\), Lemma 3 follows from Lemma 2. Suppose that, for a sum of bicompacts \(\{P_{n-1}^i\}\) satisfying the condition of Lemma 3, it has been proved that
\[ \operatorname{Ind}\left(\bigcup_{i=1}^{s} P_{n-1}^i\right)\leq n-1 . \]
We shall prove Lemma 3 for a sum of bicompacts \(\{P_n^i\}\) satisfying the condition of Lemma 3. Let
\[ P=\bigcup_{i=1}^{s} P_n^i . \]
Let \(F\subset P\) be an arbitrary closed set and \(OF\) an arbitrary neighborhood of it. We shall construct \(O_1F\subset OF\), whose boundary would be a sum of bicompacts \(\{P_{n-1}^l\}\) satisfying the condition of Lemma 3.

1. Consider \(F_1=F\cap P_n^1\) and \(OF_1=OF\cap P_n^1\). Let \(X_1^1,X_2^1,\ldots,X_n^1\) be ordered continua such that
   \[ P_n^1=\prod_{k=1}^{n} X_k^1 . \]
   Take a partition
   \[ \beta_1^1=\{\alpha_1^1\times \alpha_2^1\times\cdots\times \alpha_n^1\} \]
   of the bicompact \(P_n^1\), where \(\alpha_i^1\) \((i=1,2,\ldots,n)\) is a partition of \(X_i^1\).

The complement to the sum of those elements of the partition \(\beta_1^1\) which do not intersect \(F_1\) is an open set. Denote it by \(G_1\). The boundary of \(G_1\) is a sum of bicompacts \(\{P_{n-1}^l\}\) satisfying Lemma 3. Take \(\beta_2^1\) such that \(F_1\subseteq G_2\subseteq OF_1\).

\(k-1.\) Suppose that for \(P_n^1,\ldots,P_n^{k-1}\) there have already been constructed, respectively, such partitions
\[ \beta_k^1,\ \beta_{k-1}^2,\ldots,\beta_2^{k-1}, \]
that:

I. \(F_1\subset G_1\subset OF_1,\ldots,F_{k-1}\subset G_{k-1}\subset OF_{k-1}\) and
\[ \Gamma_{k-1}=\bigcup_{i=1}^{k-1}G_i \]
is an open set in
\[ \bigcup_{i=1}^{k-1} P_n^i . \]
The set \(G_i\) is the complement to the sum of those elements in
\[ \beta_{m+1}^{\,k-m}\quad (1\leq m\leq k-1), \]
which do not intersect \(F_i\) in \(P_n^i\).

II. If \(P_n^{i_1}\) intersects properly with \(P_n^{i_2}\) relative to \(X_k^{i_1}\), then the partitions \(\beta_{k+1-i_1}^{i_1}\) and \(\beta_{k+1-i_2}^{i_2}\) in intersection with \(X_{k(j_i)}^{i_1}\) give one and the same partition on \(X_{k(j_i)}^{i_1}\).

III. The boundary of \(G_i\) \((i=1,2,\ldots,k-1)\) and \(\Gamma_{k-1}\) is a sum of bicompacts \(\{P_{n-1}^l\}\) satisfying the condition of Lemma 3.

\(k.\) Take \(P_n^k\). Let
\[ F_k=F\cap P_n^k,\qquad OF_k=OF\cap P_n^k,\qquad F_{12\ldots k}=F\cap\left(\bigcup_{i=1}^{k}P_n^i\right)=\bigcup_{i=1}^{k}F_i, \]
\[ OF_{12\ldots k}=OF\cap\left(\bigcup_{i=1}^{k}P_n^i\right)=\bigcup_{i=1}^{k}OF_i . \]
Take on \(P_n^k\) such a partition
\[ \beta_1^k=\{\alpha_1^k\times\cdots\times\alpha_n^k\}, \]
that \(G_{k(1)}\subseteq OF_k\) (see the definition of \(G_1\) in 1). Let
\[ P_n^k=\prod_{i=1}^{n}X_i^k . \]
Take \(X_l^k\). Consider the collection \(\{P_n^{j_l}\}\) \((l\leq k)\) of bicompacts \((P_n^{j_l}=P_n^k)\), which is proper relative to \(X_l^k\) and maximal in the sense of properness. Then on \(X_{l(j_i)}^k\) two partitions are defined: \(\beta_1^k\cap X_{l(j_i)}^k\) and the one that was constructed on it at the \((k-1)\)-st step; take their intersection and denote it by \(\alpha_l^{k(1)}\). Then on the bicompacts \(P_n^1,\ldots,P_n^{k-1}\) we again obtain partitions satisfying I, II, III from item \(k-1\), if on each \(P_n^{j_l}\) we take, on the corresponding factor, the partition \(\alpha_l^{k(1)}\).

The partition \(\{\alpha_1^{k(1)} \times \alpha_2^{k(1)} \times \cdots \times \alpha_n^{k(1)}\}\) on \(P_n^k\) we shall denote by \(\beta_2^k\), and the partitions on \(P_n^1,\ldots,P_n^{k-1}\), respectively, by \(\beta_{k+1}^1,\ldots,\beta_3^{k-1}\). Then the partitions \(\beta_{k+1}^1,\ldots,\beta_2^k\) satisfy conditions I, II, III of item \(k-1\) with \(k-1\) replaced by \(k\). For \(k=n\) we obtain \(F_{12\ldots n}=F'=\Gamma_n\subset OF_{12\ldots n}=OF\), and the boundary \(\Gamma_n\) consists of the sum of bicompacts \(\{P_{n-1}^l\}\) satisfying the condition of Lemma 3. Thus, Lemma 3 is proved.

Proof of assertion 2. Let \(F\subset P_n\) be an arbitrary closed set and let \(OF\) be an arbitrary neighborhood of it. Take such a partition \(\beta=\{\alpha_1\times\cdots\times\alpha_n\}\) that \(G\subset OF\), where \(G\) is the complement of the sum of those elements of the partition \(\beta\) which do not meet \(F\). The boundary of \(G\) is the sum of bicompacts \(\{P_{n-1}^l\}\) satisfying the condition of Lemma 3, i.e. \(\operatorname{Ind} P_n\le n\).

For any bicompact \(X\), \(\dim X\le \operatorname{Ind} X\) (1). From assertions 1 and 2 it follows that for \(P_n\) we have \(\dim P_n\ge \operatorname{Ind} P_n\). Consequently, \(\dim P_n=\operatorname{ind} P_n=\operatorname{Ind} P_n=n\). Theorem 1 is proved.

Note added in proof. Already after submitting the present article for publication I managed to prove, considerably more simply, a more general result:

Theorem. Let
\[ P=\prod_{i=1}^n X_i, \]
where \(X_i\) is a bicompact \((i=1,2,\ldots,n)\) and \(\operatorname{Ind} X_i=1\). Then \(\dim P=\operatorname{ind} P=\operatorname{Ind} P=n\).

An exposition of this result will be published in this same journal.

I express my gratitude to my advisor V. I. Ponomarev for posing the problem and for his assistance.

Moscow State University
named after M. V. Lomonosov

Received
1 XI 1966

REFERENCES

1. P. S. Aleksandrov, Math. Ann., 106 (1932).
2. W. Hurewicz, H. Wallman, Dimension Theory, Moscow, 1948.