UDC 513.83

MATHEMATICS

I. K. LIFANOV

ON THE DIMENSION OF PRODUCTS OF BICOMPACTA

(Presented by Academician P. S. Aleksandrov on 27 II 1968)

§ 1. In the present section only finite covers and only normal spaces are considered. By the dimension of a space \(X\) we mean the large inductive dimension, denoted by \(\operatorname{Ind} X\). Let us recall the definition of this dimension: \(\operatorname{Ind} X \le n\) if, for every closed set \(F \subseteq X\) and every neighborhood \(OF\) of it, there exists a neighborhood \(V\) of the set \(F\) such that \(V \subseteq OF\) and \(\operatorname{Ind}([V]\setminus V) \le n-1\). By definition \(\operatorname{Ind} X=-1\) if the space \(X\) is empty.

Definition 1. For a space \(X\), a closed set \(F_1\) is called a boundary if \(F_1\) is the boundary of a set open in \(X\).

Definition 2. A space \(X\) satisfies the condition of extension of refinements*, if for any closed sets \(F_1\) and \(F_2\), \(F_1 \supseteq F_2\), \(\operatorname{Ind} F_1 > \operatorname{Ind} F_2\), \(F_2\) a boundary in \(F_1\), the following holds: for every open cover \(\omega\) of the set \(F_1\) and every refinement \(\alpha\) of the set \(F_2\), inscribed in the cover \(\omega \wedge F_2\)** with \(\operatorname{Ind} \operatorname{gr}\alpha \le \operatorname{Ind} F_2 - 1\), there exists a refinement \(\beta\) of the set \(F_1\), inscribed in the cover \(\omega\), such that \(\beta \wedge F_2=\alpha\), \(\operatorname{gr}\beta \cap F_2=\operatorname{gr}\alpha\), and \(\operatorname{Ind} \operatorname{gr}\beta \le \operatorname{Ind} F_2 - 1\).

Definition 3. We shall say that a bicompactum \(X\) belongs to the class of bicompacta \(Б\) if the following conditions are fulfilled for \(X\):

I (weakened sum theorem). For any two closed sets \(F_1\) and \(F_2\) that are boundaries,

\[ \operatorname{Ind}(F_1 \cup F_2)=\max(\operatorname{Ind}F_1,\operatorname{Ind}F_2). \]

II. The bicompactum \(X\) satisfies the condition of extension of refinements.

It turns out that for bicompacta \(X\) belonging to the class \(Б\), the following is true.

Theorem 1. Let

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

where \(X_i \in Б\) \((i=1,2,\ldots,s)\). Then

\[ \operatorname{Ind} P \le \operatorname{Ind} X_1+\ldots+\operatorname{Ind} X_s. \]

It is not hard to show that the class \(Б\) of bicompacta includes the following classes of bicompacta: a) all bicompacta whose large inductive dimension does not exceed one; b) all bicompacta that are Dowker spaces*** (see \((^2)\)), and consequently also all perfectly normal bicompacta.

The proof of part a) is contained in \((^4)\).

Corollary 1. Let \(P=X\times Y\), where \(X\) is a one-dimensional bicompactum and \(Y\) is a perfectly normal bicompactum for which \(\dim Y=\operatorname{ind}Y=\operatorname{Ind}Y=n\). Then \(\dim P=\operatorname{ind}P=\operatorname{Ind}P=n+1\).

Proof. \(\dim P \ge n+1\) follows from \((^5)\), and \(\operatorname{Ind}P \le n+1\) follows from Theorem 1.

* A cover \(\alpha=\{[O_i]\}\) \((i=1,\ldots,n)\) is called a refinement if its elements are the closures of pairwise disjoint open sets \(O_i\), called the kernels of the elements of the refinement,

\[ \operatorname{gr}\alpha=\bigcup_{j=1}^{n}[O_i]\setminus O_i. \]

** Let \(F\subseteq X\): 1) let \(\omega=\{u_1,\ldots,u_s\}\) be an open cover of the space \(X\); then \(\omega\wedge F=\{F\cap u_1,\ldots,F\cap u_s\}\); 2) let \(\alpha=\{[O_1],\ldots,[O_s]\}\) be a refinement of \(X\); then \(\alpha\wedge F=\{[O_1\cap F],\ldots,[O_n\cap F]\}\).

*** A hereditarily normal space \(X\) is called a Dowker space if every open set \(U\) is covered by a point-finite system of open sets \(F_\sigma\) in \(X\).

The proof of Theorem 1 proceeds as follows:

1) It is shown that in any open covering $\omega$ of the bicompactum $P$ one can inscribe a refinement $\alpha^1$ such that $\operatorname{bd}\alpha^1$ is the sum of a finite number of summands of the form $\prod_{i=1}^s F_i$, where $F_i$ is the boundary of some order in $X_i$, and

\[ \sum_{i=1}^s \operatorname{Ind} F_i = \sum_{i=1}^s \operatorname{Ind} X_i - 1 \]

for each summand of the set $\operatorname{bd}\alpha^1$.

2) Then it is shown that in any open covering $\omega^1$ of the set $\operatorname{bd}\alpha^1$ one can inscribe a refinement $\alpha^2$ of the set $\operatorname{bd}\alpha^1$ such that $\operatorname{bd}\alpha^2$ (relative to $\operatorname{bd}\alpha^1$) is the sum of a finite number of closed summands of the form $\prod_{i=1}^s F_i'$, and, for each summand of the set $\operatorname{bd}\alpha^2$, we have

\[ \sum_{i=1}^s \operatorname{Ind} F_i' = \sum_{i=1}^s \operatorname{Ind} X_i - 2 \quad (F_i' \subseteq X_i'). \]

After this procedure has been applied $\sum_{i=1}^s \operatorname{Ind} X_i = l$ times, we obtain that $\operatorname{bd}\alpha^l$ is the sum of a finite number of closed summands of the form $\prod_{i=1}^s F_i$, and

\[ \sum_{i=1}^s \operatorname{Ind} F_i = 0. \]

Consequently, the dimension of each set $F_i$ is zero; therefore the dimension of each summand of the set $\operatorname{bd}\alpha^l$ and the dimension of the set $\operatorname{bd}\alpha^l$ itself is zero. Then, going back, we obtain that

\[ \operatorname{Ind}\operatorname{bd}\alpha \leq \sum_{i=1}^s \operatorname{Ind} X_i - 1 \]

and, consequently,

\[ \operatorname{Ind} P \leq \sum_{i=1}^s \operatorname{Ind} X_i. \]

Definition 4. We shall say that a bicompactum $X$ belongs to the class of bicompacta $A$ if $X$ is a closed subset of some bicompactum $P$ which is the topological product of a finite number of bicompacta from the class $Б$.

Remark 1. The class $A$ is closed with respect to topological product.

Remark 2. Let $B=\prod_{i=1}^k A_i$, where $A_i \in A$ $(i=1,\ldots,k)$ and each bicompactum $A_i$ is finite-dimensional. Then the bicompactum $B$ is also finite-dimensional.

§ 2. In this section we consider the question of the dimension of the product of an infinite number of arbitrary bicompacta $X$ with $\dim X \geq 1$.

The following notion of a closed partition between disjoint closed sets is known. Let $C_1$ and $C_2$ be closed sets of a space $X$, $C_1 \cap C_2 = \varnothing$. Then a closed set $B$ is called a partition between $C_1$ and $C_2$ if $X \setminus B = V^1 \cup V^2$, $V^1 \cap V^2 = \varnothing$, the sets $V^1, V^2$ are open, and $C_1 \subseteq V^1$, $C_2 \subseteq V^2$.

P. S. Aleksandrov gave the following definition: a space $X$ is strongly infinite-dimensional if there exists a sequence of pairs $(C_i^1, C_i^2)$, $C_i^1 \cap C_i^2 = \varnothing$ $(i=1,2,\ldots)$, of closed sets such that the intersection of any closed sets $B_i$ $(i=1,2,\ldots)$, each of which is a partition for the corresponding pair $(C_i^1, C_i^2)$, is nonempty. If the space $X$ is strongly infinite-dimensional in the sense of P. S. Aleksandrov, then we shall say that it is $A$-strongly infinite-dimensional.

Now we introduce a strengthening of the notion of partition.

Definition 5. Let there be given closed sets $C^1$ and $C^2$ in $X$, with $C^1 \cap C^2 = \varnothing$. A set $B$ is called a thick partition between $C^1$ and $C^2$ if $X \setminus \operatorname{Int} B = \Phi_1 \cup \Phi_2$, $\Phi_1 \cap \Phi_2 = \varnothing$, the sets $\Phi_1, \Phi_2$ are closed, $C^i \subseteq \operatorname{Int}\Phi_i$ $(i=1,2)$ ($\operatorname{Int} B$ denotes the interior of the set $B$), and $B \cap (C^1 \cup C^2)=\varnothing$.

If the space \(X\) is normal, then the following assertions are equivalent:

I. In the space \(X\) there exist \(n\) pairs of closed sets \((C_i^1, C_i^2)\) \((i=1,2,\ldots,n)\), \(C_i^1\cap C_i^2=\varnothing\), such that the intersection of any closed sets \(B_i\) \((i=1,2,\ldots,n)\), each of which is a thick partition for the pair \((C_i^1,C_i^2)\), is nonempty.

II. In the space \(X\) there exist \(n\) pairs of closed sets \((C_i^1, C_i^2)\) \((i=1,2,\ldots,n)\), \(C_i^1\cap C_i^2=\varnothing\), such that the intersection of arbitrary closed sets \(B_i\) \((i=1,2,\ldots,n)\), each of which is a partition for the pair \((C_i^1,C_i^2)\), is nonempty.

Definition 6. A normal space \(X\) will be called \(L\)-strongly infinite-dimensional if there exists a sequence of pairs \((C_i^1, C_i^2)\) \((i=1,2,\ldots)\), \(C_i^1\cap C_i^2=\varnothing\), of closed sets such that the intersection of arbitrary closed sets \(B_i\) \((i=1,2,\ldots)\), each of which is a thick partition for the corresponding pair \((C_i^1,C_i^2)\), is nonempty.

Lemma 1. If the space \(X\) is a bicompactum, then the definitions of \(L\)-strong infinite-dimensionality and \(A\)-strong infinite-dimensionality are equivalent.

Proof. a) From \(A\)-strong infinite-dimensionality, obviously, \(L\)-strong infinite-dimensionality follows.

b) Let the bicompactum \(X\) be \(L\)-strongly infinite-dimensional. Suppose that it is not \(A\)-strongly infinite-dimensional. Then for any sequence of pairs of closed sets \((F_i^1,F_i^2)\), \(F_i^1\cap F_i^2=\varnothing\) \((i=1,2,\ldots)\), there exists a sequence of closed sets \(A_i\) \((i=1,2,\ldots)\) (each of which is a partition respectively for the pair \((F_i^1,F_i^2)\)) such that

\[ \bigcap_{i=1}^{\infty} A_i=\varnothing. \]

Consequently, also for the sequence of pairs of closed sets \((C_i^1,C_i^2)\) participating in the definition of \(L\)-strong infinite-dimensionality, there exists a corresponding sequence of closed sets \(A_i\) \((i=1,2,\ldots)\). Since \(X\) is a bicompactum and

\[ \bigcap_{i=1}^{\infty} A_i=\varnothing, \]

there exists an integer \(k\) such that

\[ \bigcap_{i=1}^{k} A_i=\varnothing. \]

Therefore one can take open sets \(O_i\) \((i=1,2,\ldots,k)\) such that \(A_i\subseteq O_i\), \([O_i]\cap(C_i^1\cup C_i^2)=\varnothing\), and

\[ \bigcap_{i=1}^{k} [O_i]=\varnothing. \]

Now, taking for each \(j>k\) an arbitrary open set \(O_j\) such that \(A_j\subseteq O_j\) and \([O_j]\cap(C_j^1\cap C_j^2)=\varnothing\), we obtain such a sequence of closed sets \([O_i]\) \((i=1,2,\ldots)\) (which are thick partitions for the corresponding pairs \((C_i^1,C_i^2)\)) that

\[ \bigcap_{i=1}^{\infty} [O_i]=\varnothing. \]

We have arrived at a contradiction with the choice of the sequence of pairs of closed sets \((C_i^1,C_i^2)\) \((i=1,2,\ldots)\).

Remark 3. Lemma 1 is also true for countably paracompact spaces (spaces in every countable open cover of which one can inscribe a locally finite cover (see \((^6)\))).

Theorem 2. Let \(P=\prod_{\alpha\in A} X_\alpha\), where \(X_\alpha\) are bicompacta, \(\dim X_\alpha\geq 1\) for every \(\alpha\in A\), and \(m(A)\geq \aleph_0\). Then the bicompactum \(P\) is \(L\)-strongly infinite-dimensional.

Proof. It is enough to consider the case when

\[ P=\prod_{i=1}^{\infty} X_i \]

is the topological product of a countable number of bicompacta \(X_i\). Since \(\dim X_i\geq 1\), in the bicompactum \(X_i\) there exist points \(x_i^1\ne x_i^2\) which cannot be separated by the empty set. Take the sets

\[ C_i^1=\prod_{j\ne i} X_j\times x_i^1,\qquad C_i^2=\prod_{j\ne i} X_j\times x_i^2. \]

Then \(C_i^1\) and \(C_i^2\) are closed in the bicompactum \(P\), \(C_i^1\cap C_i^2=\varnothing\), and it remains only to prove that the intersection of arbitrary closed sets \(B_i\) \((i=1,2,\ldots)\), each of which

is a thick partition for the pair \((C_i^1,C_i^2)\), respectively, is nonempty. To prove the last assertion, we note that for this it is sufficient to prove the nonemptiness of the intersection of any finite number of the sets \(B_i\), since then the assertion we need will follow from the bicompactness of the space \(P\). Thus, the proof of Theorem 2 has been reduced to the proof of the following assertion.

Let
\[ P_n=\prod_{i=1}^{n}X_i, \]
where the \(X_i\) are bicompacts and \(\dim X_i\geqslant 1\) for every \(i=1,2,\ldots,n\). Then the sequence of pairs of closed sets \((C_i^1,C_i^2)\) (where
\[ C_i^1=\prod_{j\ne i}^{n}X_j\times x_i^1,\qquad C_i^2=\prod_{j\ne i}^{n}X_j\times x_i^2 \]
) is such that the intersection of arbitrary closed sets \(B_i\) \((i=1,2,\ldots,n)\), each of which is a thick partition for the pair \((C_i^1,C_i^2)\), is nonempty.

We shall prove the last assertion for the case \(n=2\). Let
\[ P_2=\prod_{i=1}^{2}X_i,\qquad \dim X_i\geqslant 1\quad (i=1,2), \]
and let \(B_1,B_2\) be thick partitions for the pairs of closed sets \((C_1^1,C_1^2)\) and \((C_2^1,C_2^2)\), respectively. Suppose that \(B_1\cap B_2=\varnothing\). Then the sets
\[ F_i^1=B_i\cap \Phi_i^1,\qquad F_i^2=B_i\cap \Phi_i^2 \]
(where \(\Phi_i^1\cup\Phi_i^2=P_2\setminus \operatorname{Int}B_i\) \((i=1,2)\) and \(\Phi_i^1\cap\Phi_i^2=\varnothing\)) are pairwise disjoint, i.e.
\[ F_i^1\cap F_i^2=\varnothing \]
for \(i=1,2\). For fixed \(i\), the sets \(F_i^1,F_i^2,C_i^1,C_i^2\) are also pairwise disjoint. Therefore one can take an open cover \(\omega\) of the bicompact \(P_2\) such that:

\[ (*)\quad \]
no element of this cover intersects simultaneously an arbitrary pair of disjoint sets from the system
\[ F_i^1,\ F_i^2,\ B_i,\ \Phi_i^1,\ \Phi_i^2,\ C_i^1,\ C_i^2\quad (i=1,2). \]

We refine this cover \(\omega\) by an open cover
\[ \widetilde{\omega}=\{\omega_1\times\omega_2\}, \]
whose elements are sets of the form \((u_i^1\times u_j^2)\), where \(u_i^1\in\omega_1\), \(u_j^2\in\omega_2\), and \(\omega_1,\omega_2\) are open covers of the bicompacts \(X_1,X_2\), respectively. Let \(\pi_i:P_2\to X_i\) be the natural projection of \(P_2\) onto the factor \(X_i\). Then one may require that no element of the cover \(\omega_i\) intersect a pair of sets \((x_i^1,\pi_i(B_i))\) and \((x_i^2,\pi_i(B_i))\). Now take, for each \(\omega_i\), an \(\omega_i\)-map \(f\) into the body of the nerve \(\widetilde K_i\) of the cover (see \((2)\)). Then the points \(f_i(x_i^1)\) and \(f_i(x_i^2)\) lie in one component of the polyhedron \(\widetilde K_i\), and therefore there exists a segment \(I_i\) whose initial point is \(f_i(x_i^1)\) and whose endpoint is \(f_i(x_i^2)\). Let
\[ X_i^+=f_i^{-1}(I_i). \]
The set \(X_i^+\) is closed in \(X_i\), and \(x_i^j\in X_i^+\) \((j=1,2)\). The mapping
\[ f=f_1\times f_2 \]
of the bicompact \(P_2\) maps the set
\[ P_2^+=X_1^+\times X_2^+ \]
onto a square, and the sets
\[ P_2^+\cap C_i^j=+C_i^j=x_i^j\times X_k^+\quad (i=1,2;\ i=k;\ k=1,2;\ j=1,2) \]
are mapped onto opposite sides of the square for fixed \(i\). Let
\[ \omega^+=P_2^+\wedge\omega; \]
then
\[ f^+=f/P_2^+, \]
the restriction of the mapping \(f\) to the set \(P_2^+\), will be an \(\omega^+\)-map, and
\[ B_i^+=B_i\cap P_2^+. \]
The set \(f^+(B_i^+)\) separates the sets \(f^+(+C_i^1)\) and \(f^+(+C_i^2)\), since \(f^+\) is an \(\omega^+\)-map and the cover \(\omega\) of the bicompact \(P_2\) satisfies condition \((*)\). Consequently,
\[ f^+(B_1^+)\cap f^+(B_2^+)=\varnothing, \]
and this contradicts the fact that
\[ B_1^+\cap B_2^+=\varnothing, \]
\(f^+\) is an \(\omega^+\)-map, and no element of the cover \(\omega^+\) intersects simultaneously the sets \(B_1^+\) and \(B_2^+\). Consequently,
\[ B_1^+\cap B_2^+=P_2^+\cap(B_1\cap B_2)\ne\varnothing, \]
i.e.
\[ B_1\cap B_2\ne\varnothing. \]
Theorem 2 is proved.

In conclusion I express my heartfelt gratitude to B. A. Pasynkov for his constant attention and help.

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

Received
23 II 1968

References

1. P. S. Alexandroff, Proc. Roy. Soc., A 189, No. 2 (1947).
2. P. S. Alexandroff, Combinatorial Topology, 1947.
3. B. A. Pasynkov, DAN, 175, No. 2 (1967).
4. I. K. Lifanov, DAN, 180, No. 3 (1968).
5. H. Cohen, Duke Math. J., 21, No. 2, 209 (1954).
6. C. H. Dowker, Canad. J. Math., 3, No. 2, 219 (1951).