UDC 513.83

MATHEMATICS

I. K. LIFANOV

ON THE DIMENSION OF THE PRODUCT OF ONE-DIMENSIONAL BICOMPACTA

(Presented by Academician P. S. Aleksandrov on 13 VI 1967)

In this paper we shall prove the following two theorems:

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

Theorem 2. Let
\[ P=\prod_{i=1}^{\infty} X_i, \]
where \(X_i\) is an ordered continuum for every \(i\). Then \(P\) is strongly infinite-dimensional.

Before proving the theorems, we recall some known definitions and formulate several lemmas.

Definition 1. Let \(X\) be a normal space. A cover \(\alpha=\{A_1,\ldots,A_n\}\) of the space \(X\) is called a partition if \(A_i=[\operatorname{Int} A_i]\) and \(\operatorname{Int} A_i\cap \operatorname{Int} A_j=\varnothing\) for \(i\ne j\).

We introduce a new definition.

Definition 2. Let \(\alpha=\{A_1,\ldots,A_n\}\) be a partition of the space \(X\); then by \(\operatorname{bd}\alpha\) we shall denote the set
\[ X\setminus \bigcup_{i=1}^{n}\operatorname{Int} A_i . \]

Lemma 1. Let \(X\) be a normal space and \(\operatorname{Ind} X\le 1\). Let \(F\subset X\) be a closed set, \(\operatorname{Ind} F\le 0\), and let \(\gamma=\{\Gamma_1,\ldots,\Gamma_n\}\) be a cover of \(F\) by sets open in it. Then into any open cover \(\omega=\{O_1,\ldots,O_m\}\) of the space \(X\) one can inscribe a partition \(\alpha=\{A_1,\ldots,A_l\}\) such that: 1) \(\operatorname{Ind}\operatorname{bd}\alpha\le 0\) and \(\operatorname{bd}\alpha\cap F=\varnothing\); 2) \(\alpha\wedge F\) is a partition of \(F\), consisting of sets open-and-closed in \(F\) and inscribed in the cover \(\gamma\).

Lemma 2. Let \(X\) be a normal space. If into every finite open cover of the space \(X\) one can inscribe a partition \(\alpha\) such that \(\operatorname{Ind}\operatorname{bd}\alpha\le n-1\), then \(\operatorname{Ind}X\le n\).

Proof of Theorem 1.

A. We first prove the inequality \(\operatorname{Ind}P\le n\).

The proof will be by induction. We shall show that 1) into any cover \(\omega\) of the bicompactum \(P\) one can inscribe a partition \(\alpha^1\), such that 2) into any cover of \(\operatorname{bd}\alpha^1\) one can inscribe a partition \(\alpha^2\) of the set \(\operatorname{bd}\alpha^1\), with \(\operatorname{bd}\alpha^2\) such that, 3) into any … and so on, that \(n-1\)) into any cover of \(\operatorname{bd}\alpha^{n-1}\) one can inscribe a partition \(\alpha^n\) such that \(\operatorname{Ind}\operatorname{bd}\alpha^n\le 0\). Then we shall obtain that \(\operatorname{Ind}\operatorname{bd}\alpha^{n-1}\le 1\), \(\operatorname{Ind}\operatorname{bd}\alpha^1\le n-1\), and \(\operatorname{Ind}P\le n\).

1) Let \(\omega^1=\{O_1^1,\ldots,O_s^1\}\) be an arbitrary open cover of the space \(P\). Then into \(\omega^1\) one can inscribe a cover
\[ \widetilde{\omega}^{\,1}=\{\omega_1^1\times\cdots\times\omega_n^1\}, \]
where \(\omega_i^1\) is a cover of the space \(X_i\), and \(\widetilde{\omega}^{\,1}\) is a family of open sets, each of which is the product of \(n\) sets, one from each \(\omega_i^1\) \((i=1,2,\ldots,n)\). Into each cover \(\omega_i^1\) we inscribe a partition \(\alpha_i^1\) such that \(\operatorname{Ind}\operatorname{bd}\alpha_i^1\le 0\); then the partition
\[ \alpha^1=\{\alpha_1^1\times\cdots\times\alpha_n^1\} \]
will be inscribed in the cover \(\widetilde{\omega}^{\,1}\), and
\[ \operatorname{bd}\alpha^1= \bigcup_{i=1}^{n}\prod_{\substack{i=1\\ i\ne i_1}}^{n} X_i\times \operatorname{bd}\alpha_{i_1}^1 . \]

2) Let now \(\omega^2=\{O_1^2,\ldots,O_{i_{\omega^3}}^2\}\) be an arbitrary covering of \(\operatorname{Fr}\alpha^1\). Denote

\[ X^{i_1}=\prod_{\substack{i=1\\ i\ne i_1}}^n X_i\times \operatorname{Fr}\alpha_{i_1}^1. \]

On each \(X^l\) \((l=1,2,\ldots,n)\) inscribe, in the covering \(X^l\wedge \omega^2\), the covering \(\gamma_l=\{\gamma_1^l\times \ldots\times \gamma_n^l\}\). Then on \(X_i\) we shall have \((n-1)\) coverings and one covering on \(\operatorname{Fr}\alpha_i^1\).

Take on \(X_i\) such a covering \(\widetilde\gamma_i\) that it is inscribed in all the coverings on \(X_i\), and \(\widetilde\gamma_i\wedge \operatorname{Fr}\alpha_i^1\) is inscribed in the covering \(\gamma_i^i\). Now, by Lemma 2, inscribe in the covering \(\widetilde\gamma_i\) such a partition \(\widetilde\alpha_i^2\) that \(\widetilde\alpha_i^2\) is inscribed in the covering \(\widetilde\gamma_i\) and a) \(\operatorname{Ind}\operatorname{Fr}\widetilde\alpha_i^2\leqslant 0\), \(\operatorname{Fr}\widetilde\alpha_i^2\cap \operatorname{Fr}\alpha_i^1=\varnothing\); b) \(\widetilde\alpha_i^2\wedge \operatorname{Fr}\alpha_i^1\) is a partition into sets open-and-closed in \(\operatorname{Fr}\alpha_i^1\), which we denote by \(\beta_i\). Take now on \(X_i\) the partition \(\alpha_i^2=\widetilde\alpha_i^2\wedge \alpha_i^1\). If on each \(X^l\) we take the partition

\[ \widetilde\alpha^l= \prod_{\substack{i=1\\ i\ne l}}^n \alpha_i^2\times \beta_l, \]

then

\[ \alpha^2=\bigvee_{l=1}^n \widetilde\alpha^l \]

(where the sign \(\bigvee_{l=1}^n\) denotes the sum in the sense of taking the whole collection of elements of the partitions \(\widetilde\alpha^l\), and not in the sense of the union of sets from \(\widetilde\alpha^l\)) will be a partition inscribed in the covering \(\omega^2\).

Let us prove this last assertion. Indeed:

a) the inscription of the system of sets \(\alpha^2\) in the covering \(\omega^2\) is fulfilled by construction.

b) Let \(A\) be an arbitrary element of some \(\widetilde\alpha^l\); then \(\operatorname{Int}_{X^l} A\) contains points from \(\operatorname{Fr}\alpha_l^1\), but contains no points from \(\operatorname{Fr}\alpha_i^1\) \((i\ne l,\ i=1,2,\ldots,n)\). Therefore

\[ \operatorname{Int}_{X^l} A\cap X^i=\varnothing \quad (i\ne l,\ i=1,2,\ldots,n), \]

since \(X^i\) contains the factor \(\operatorname{Fr}\alpha_i^1\) \((i\ne l,\ i=1,2,\ldots,n)\). Consequently, the \(\operatorname{Int}_{X^l} A\) are open in \(\operatorname{Fr}\alpha^1\), and these sets are pairwise disjoint; the sum of their closures will give the entire set \(\operatorname{Fr}\alpha^1\).

The assertion is proved. Thus, \(\alpha^2\) is a partition of \(\operatorname{Fr}\alpha^1\), inscribed in the covering \(\omega^2\), and

\[ \operatorname{Fr}\alpha^2= \bigcup_{i_1=1}^n \bigcup_{\substack{i_2=1\\ i_2\ne i_1}}^n \prod_{\substack{i=1\\ i\ne i_1,i_2}}^n X_i\times \operatorname{Fr}\alpha_{i_1}^1\times \operatorname{Fr}\alpha_{i_2}^2. \]

Suppose that at the \((k-1)\)-st step we already have

\[ \operatorname{Fr}\alpha^{k-1}= \bigcup_{i_1=1}^n \bigcup_{\substack{i_2=1\\ i_2\ne i_1}}^n \cdots \bigcup_{\substack{i_{k-1}=1\\ i_{k-1}\ne i_1,\ldots,i_{k-2}}}^n \prod_{\substack{i=1\\ i\ne i_1,\ldots,i_{k-1}}}^n X_i\times \operatorname{Fr}\alpha_{i_1}^1\times \operatorname{Fr}\alpha_{i_2}^2\times \cdots \]

\[ \cdots \times \operatorname{Fr}\alpha_{i_{k-1}}^{k-1}, \]

where

\[ \operatorname{Fr}\alpha_i^1\subset \operatorname{Fr}\alpha_i^2\subset \cdots\subset \operatorname{Fr}\alpha_i^{k-1} \]

and \(\operatorname{Fr}\alpha_i^l\) is open-and-closed in \(\operatorname{Fr}\alpha_i^{l+1}\), \(1\leqslant l\leqslant k-2\).

Denote

\[ X^{i_1 i_2\ldots i_{k-1}} = \bigcup_{P(1,2,\ldots,k-1)} \prod_{\substack{i=1\\ i\ne i_1,\ldots,i_{k-1}}}^n X_i\times \operatorname{Fr}\alpha_{i_1}^{j_1}\times \cdots\times \operatorname{Fr}\alpha_{i_{k-1}}^{j_{k-1}} \]

\[ (i\ne i_1,\ldots,i_{k-1};\ (j_1,\ldots,j_{k-1})\text{ is an element of the permutation of }1,2,\ldots,k-1), \]

\[ X_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}} = \prod_{\substack{i=1\\ i\ne i_1,\ldots,i_{k-1}}}^n X_i\times \operatorname{Fr}\alpha_{i_1}^{j_1}\times \cdots\times \operatorname{Fr}\alpha_{i_{k-1}}^{j_{k-1}}, \]

then

\[ X^{i_1 i_2\ldots i_{k-1}} = \bigcup_{P(1,2,\ldots,k-1)} X_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}}, \qquad \operatorname{Fr}\alpha^{k-1} = \bigcup_{C_n^{\,i_1\ldots i_{k-1}}} X^{i_1 i_2\ldots i_{k-1}}. \]

We carry out the \(k\)-th step. Let \(\omega^k\) be an arbitrary cover by open sets of the set \(\operatorname{gr}\alpha^{k-1}\). On each \(X_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}}\) we inscribe in the cover
\(\omega^k \wedge X_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}}\) a cover
\(\gamma_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}}=\{\gamma_1\times\cdots\times\gamma_n\}\).
Then on \(X_i\) we obtain a certain number of covers both on
\(\operatorname{gr}\alpha_i^1,\operatorname{gr}\alpha_i^2,\ldots,\operatorname{gr}\alpha_i^{k-1}\).
Take on \(X_i\) a cover \(\widetilde{\gamma}_i\) by open sets such that it is inscribed in all the given covers on \(X_i\), and
\(\widetilde{\gamma}_i\wedge \operatorname{gr}\alpha_i^j\) \((j=1,2,\ldots,k-1)\) is inscribed in all the given covers on \(\operatorname{gr}\alpha_i^j\).

Now, by Lemma 2, inscribe in the cover \(\widetilde{\gamma}_i\) such a partition \(\widetilde{\alpha}_i^k\) that \(\widetilde{\alpha}_i^k\) is inscribed in \(\widetilde{\gamma}_i\) and a) \(\operatorname{Ind}\operatorname{gr}\widetilde{\alpha}_i^k\leq 0\),
\(\operatorname{gr}\widetilde{\alpha}_i^k\cap \operatorname{gr}\alpha_i^{k-1}\ne\varnothing\);
b) \(\widetilde{\alpha}_i^k\wedge \operatorname{gr}\alpha_i^{k-1}\) is a partition of \(\operatorname{gr}\alpha_i^{k-1}\) into sets open-and-closed in \(\operatorname{gr}\alpha_i^{k-1}\), which we denote by \(\beta_i^{k-1}\); moreover, we require that, if an element of the partition \(\beta_i^{k-1}\) meets \(\operatorname{gr}\alpha_i^j\) \((1\leq j\leq k-1)\), then it is contained in it, and we denote
\(\beta_i^j=\beta_i^{k-1}\wedge \operatorname{gr}\alpha_i^j\) \((1\leq j\leq k-1)\).
Next take the partition \(\alpha_i^k=\widetilde{\alpha}_i^k\wedge \alpha_i^{k-1}\); then on each
\(X_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}}\) we obtain the partition

\[ \alpha_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}} = \{\alpha_1^k\times\cdots\times \alpha_{i_1-1}^k\times \beta_{i_1}^{j_1}\times \alpha_{i_1+1}^k\times\cdots\times \alpha_{i_{k-1}-1}^k\times \beta_{i_{k-1}}^{j_{k-1}}\times \alpha_{i_{k-1}+1}^k\times\cdots \]
\[ \cdots\times \alpha_n^k\}. \]

If an element of the partition
\(\alpha_{j_1\ldots j_{k-1}}^{i_1\ldots k-1}\) meets some bicompactum among the constituent bicompacta
\(X^{i_1 i_2\ldots i_{k-1}}\), then it belongs entirely to this bicompactum and is an element of a partition on it.

Thus, if one takes

\[ \alpha^{i_1\ldots i_{k-1}} = \bigvee_{P(1,\ldots,k-1)} \alpha_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}}, \]

where the sum is understood in the sense that, if two elements from different summands meet, then they are identified as one. Then
\(\alpha^{i_1\ldots i_{k-1}}\) is a partition of the bicompactum
\(X^{i_1\ldots i_{k-1}}\) by virtue of the properties described above of the partitions on
\(X_{j_1\ldots j_{k-1}}^{i_1\ldots i_{k-1}}\)
\((j_1\ldots j_{k-1}\in P(1,2,\ldots,k-1))\).
Further, if one takes

\[ \alpha^k=\bigvee_{C_n^{\,i_1\ldots i_{k-1}}} \alpha^{i_1\ldots i_{k-1}}, \]

then we obtain that \(\alpha^k\) is a partition on \(\operatorname{gr}\alpha^{k-1}\). Indeed, if \(A\) is an element of the partition \(\alpha^{i_1\ldots i_{k-1}}\), then
\(\operatorname{Int}A\cap X^{j_1\ldots j_{k-1}}=\varnothing\)
(for \(j_1\ldots j_{k-1}\ne i_1\ldots i_{k-1}\), i.e., if they differ in at least one index). We have

\[ \operatorname{gr}\alpha^k = \bigcup_{i=1}^{n} \bigcup_{\substack{i_2=1\\ i_2\ne i_1}}^{n} \cdots \bigcup_{\substack{i_k=1\\ i_k\ne i_1,\ldots,i_{k-1}}}^{n} \prod_{i=1}^{n} X_i \times \operatorname{gr}\alpha_{i_1}^{k} \times \cdots \times \operatorname{gr}\alpha_{i_k}^{k} \]

(where \(i\ne i_1,\ldots,i_k\)) or

\[ \operatorname{gr}\alpha^k = \bigcup_{C_n^{\,i_1\ldots i_k}} \bigcup_{P(1,2,\ldots,k)} \prod_{i=1}^{n} X_i \times \operatorname{gr}\alpha_{i_1}^{j_1} \times \cdots \times \operatorname{gr}\alpha_{i_k}^{j_k} \quad (i\ne i_1,\ldots,i_k). \]

For \(k=n\) we obtain that \(\operatorname{gr}\alpha^n\) is the sum of a finite number of zero-dimensional bicompacta. Consequently,
\(\operatorname{Ind}\operatorname{gr}\alpha^n\leq 0\), while
\(\operatorname{Ind}\operatorname{gr}\alpha^{n-1}\leq 1\), hence
\(\operatorname{Ind}\operatorname{gr}\alpha^1\leq n-1\).
Thus the inequality \(\operatorname{Ind}P\leq n\) is proved.

B. We shall now prove the inequality \(\dim P\geq n\).

This inequality for our case follows from the work of Cohen \((^2)\), where he introduces the notion of cohomological dimension, which goes back to the work of P. S. Aleksandrov \((^1)\).

Cohen denotes his dimension by \(\operatorname{cd}\) and proves in \((^2)\) the following theorems, which we shall formulate for the case when the space \(X\) is a bicompactum (we retain Cohen’s numbering).

Theorem 7.2. For any bicompactum \(X\), \(\operatorname{cd}(X) \leq \dim X\).

Theorem 5.8. If \(X\) is a bicompactum and \(\operatorname{ind} X = 1\), then \(\operatorname{cd}(X)=1\).

Theorem 6.5. If \(\operatorname{cd}(X)=n\) and \(\operatorname{cd}(Y)=1\), \(X,Y\) are bicompacta, then any of the following conditions is sufficient for \(\operatorname{cd}(X\times Y)=n+1\):
a) \(n=0\), b) \(\operatorname{ind} X=n\), c) \(\operatorname{ind} Y=1\).

It follows from these theorems that \(\dim P \geq n\).

The theorem of P. S. Aleksandrov stating that, for bicompacta, \(\dim X \leq \operatorname{ind} X\), completes the proof of our Theorem 1.

Before proving our Theorem 2, let us recall the definition of a strongly infinite-dimensional space, given by P. S. Aleksandrov.

Definition. A space \(X\) is called strongly infinite-dimensional if there exists a countable number of pairs of closed sets \(\{C_i, C_i'\}\) such that \(C_i\cap C_i'=\varnothing\), and such that the intersection of any closed sets \(B_i\), each of which is a partition* for the pair \((C_i,C_i')\), is nonempty \((i=1,2,\ldots,n,\ldots)\).

Proof of Theorem 2. Every ordered bicompactum has a minimal and a maximal point.

Denote the end points of the bicompactum \(X_i\) by \(i_0\) and \(i_1\). Then, as the pairs \((C_i,C_i')\), one may take the sets

\[ C_i=\prod_{\substack{j=1\\ j\ne i}}^\infty X_j\times i_0,\qquad C_i'=\prod_{\substack{j=1\\ j\ne i}}^\infty X_j\times i_1. \]

Now let \(B_i\) be an arbitrary closed partition for the pair \((C_i,C_i')\) \((i=1,2,\ldots,n)\). We shall prove that \(\bigcap_{i=1}^n B_i\ne\varnothing\). Since \(P\) is a bicompactum, it suffices to prove that the intersection of any finite number of the sets \(B_i\) is nonempty. This follows from the proof of assertion 1 in my paper \((^5)\). Theorem 2 is proved.

Theorem 2 is a positive answer to a question posed to me by Yu. M. Smirnov during my talk at the International Congress of Mathematicians in Moscow in 1966.

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

Received
9 VI 1967

References

1. P. S. Aleksandrov, Proc. Roy. Soc. London, A 189, 11 (1947).
2. H. Cohen, Duke, Math. J., 21, No. 2, 209 (1954).
3. P. S. Aleksandrov, Soobshch. AN GruzSSR, 2, No. 1—2, 1 (1941).
4. W. Hurewicz, H. Wallman, Dimension Theory, 1948.
5. I. K. Lifanov, DAN, 177, No. 4 (1967).

* A partition in the space \(X\) between \(C\) and \(C'\) \((C\cap C'=\varnothing)\) is a set \(B\) such that \(X\setminus B=G\cup H\), where \(G\) and \(H\) are disjoint open subsets of \(X\) such that \(C\subset G\), and \(C'\subset H\).