UDC 513.831

MATHEMATICS

I. K. LIFANOV

SOME THEOREMS CONCERNING THE DIMENSION Ind

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

Dauker \((^1)\) proves sum theorems for \(\operatorname{Ind}\) * for hereditarily normal spaces. In this note we free ourselves from the condition of hereditary normality ** of the spaces participating in the sum, but impose conditions on the intersections of these spaces.

Theorem 1. If a normal space \(P=X\cup Y\) is the sum of two closed subsets \(X\) and \(Y\) such that: 1) \(\operatorname{Ind}X\le n\), \(\operatorname{Ind}Y\le n\); 2) \(\operatorname{Ind}(X\cap Y)\le 0\), then \(\operatorname{Ind}P\le n\).

Proof. Let \(F=X\cap Y\). Take a closed set \(\Phi\) in \(P\) and an arbitrary neighborhood \(O\Phi\) of it.

a) If \(\Phi\cap F=\varnothing\), then for \(O\Phi\) there will be found a neighborhood \(O_1\Phi\) of the set \(\Phi\) such that \([O_1\Phi]\cap F=\varnothing\), \(O_1\Phi\subseteq O\Phi\), and \(\operatorname{Ind}\operatorname{fr}O_1\Phi\le n-1\).

b) Let now \(\Phi\cap F\ne\varnothing\). Put \(\Phi_1=\Phi\cap F\), \(O_F\Phi_1=O\Phi\cap F\), where the subscript \(F\) indicates in which set this \(O_F\Phi_1\) is open.

Since \(\operatorname{Ind}F\le 0\), there will be found an open-and-closed neighborhood \(O_F^1\Phi_1\) of the set \(\Phi_1\), contained in \(O_F\Phi_1\).

Consider the sets \(\Phi_2=\Phi\cup O_F^1\Phi_1\) and \(\Phi_2'=(P\setminus O\Phi)\cup F_1\), where \(F_1=F\setminus O_F^1\Phi_1\). The sets \(\Phi_2\) and \(\Phi_2'\) are closed in \(P\), and \(\Phi_2\cap\Phi_2'=\varnothing\). Next take the sets \(X_1=X\cap\Phi_2\), \(X_1'=X\cap\Phi_2'\) and the sets \(Y_1=Y\cap\Phi_2\), \(Y_1'=Y\cap\Phi_2'\). The sets \(X_1, X_1'\) (\(Y_1, Y_1'\)) are closed in \(X\) (\(Y\)) and \(X_1'\cap X_1=\varnothing\) (\(Y_1\cap Y_2'=\varnothing\)); therefore, in view of the fact that \(\operatorname{Ind}X\le n\), \(\operatorname{Ind}Y\le n\), there will be found a neighborhood \(\Gamma_1\) (\(\Gamma_2\)) of the set \(X_1\) (\(Y_1\)) in \(X\) (\(Y\)) such that \(\operatorname{Ind}\operatorname{fr}\Gamma_1\le n-1\) (\(\operatorname{Ind}\operatorname{fr}\Gamma_2\le n-1\)) and \([\Gamma_1]_X\cap X_1'=\varnothing\) (\([\Gamma_2]_Y\cap Y_1'=\varnothing\)). From the choice of the sets \(\Gamma_1\) and \(\Gamma_2\) it is clear that the set \(\Gamma=\Gamma_1\cup\Gamma_2\) is open in \(P\), since \(\Gamma_1\cap Y=\Gamma_2\cap X\). Moreover, \(\operatorname{fr}_P\Gamma_1\cap\operatorname{fr}_P\Gamma_2=\varnothing\), since \(\operatorname{fr}_P\Gamma_1\subset X\), \(\operatorname{fr}_P\Gamma_2\subset Y\) and \(\operatorname{fr}_P\Gamma_1\cap F=\operatorname{fr}_P\Gamma_2\cap F=\varnothing\), while \(\operatorname{fr}_P\Gamma_1\cup\operatorname{fr}_P\Gamma_2\) is closed. Then \(\operatorname{Ind}\operatorname{fr}_P\Gamma\le n-1\) and \(\Phi\subset \Gamma\subset O\Phi\). Thus theorem 1 is proved.

From theorem 1 it follows

Theorem 2. Let a normal space
\[ P=\bigcup_{i=1}^n X_i, \]
where: 1) \(X_i\) is closed in \(P\); 2) \(\operatorname{Ind}(X_i\cap X_j)\le 0\) for \(i\ne j\) \((i=1,2,\ldots,n;\ j=1,2,\ldots,n)\); 3) \(\operatorname{Ind}X_i\le n\) \((i=1,2,\ldots,n)\). Then \(\operatorname{Ind}P\le n\).

Using theorem 2, we shall now prove theorem 3.

Theorem 3. Let \(P=X\times Y\), where \(X,Y\) are bicompacts and \(\operatorname{Ind}X\le 1\), \(\operatorname{Ind}Y\le 1\). Then \(\operatorname{Ind}P\le 2\).

We shall first prove a lemma.

Lemma 1. Let \(X\) be a one-dimensional bicompact, \(\operatorname{Ind}X\le 1\), and let \(\omega=\{O_1,O_2,\ldots,O_s\}\) be an arbitrary open covering. Then one can construct a refinement *** \(\alpha=\{A_1,A_2,\ldots,A_s\}\), inscribed in \(\omega\), satisfying the condition \(\operatorname{Ind}\operatorname{fr}(\operatorname{Int}A_j)\le 0\).

* A space \(X\) has \(\operatorname{Ind}X\le n\) if for every closed set \(F\) and every neighborhood \(OF\) of it there exists \(O_1F\subset OF\) such that \(\operatorname{Ind}\operatorname{fr}O_1F\le n-1\), \(\operatorname{Ind}\varnothing=-1\).

** All spaces considered in the note are assumed to be Hausdorff and normal.

* A covering \(\alpha=\{A_1,A_2,\ldots,A_s\}\) of a 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 \(i\ne j\) \((i,j=1,2,\ldots,n)\).

Proof. Take an arbitrary open cover
\(\omega=\{O_1,O_2,\ldots,O_s\}\). In \(\omega\) inscribe such an open cover
\(\omega^1=\{O_1^1,O_2^1,\ldots,O_s^1\}\) that \([O_i^1]\subseteq O_i\) and
\(\operatorname{Ind}\operatorname{Fr} O_i^1\leq 0,\ i=1,2,\ldots,s\).

Define by induction the sets \(A_i\) \((i=1,2,\ldots,s)\),
\[ \operatorname{Int} A_1 = O_1^1\setminus \bigcup_{i=2}^{s}[O_i^1],\qquad A_1=[\operatorname{Int} A_1], \]
then
\[ A_1\cup \bigcup_{i=2}^{s} O_i^1=X \quad\text{and}\quad \operatorname{Ind}\operatorname{Fr}(\operatorname{Int} A_1)\leq 0, \]
since
\[ \operatorname{Fr}(\operatorname{Int} A_1)\subseteq \bigcup_{i=1}^{s}\operatorname{Fr} O_i^1 . \]

Suppose that we have already constructed sets \(A_1,A_2,\ldots,A_{k-1}\) satisfying the conditions:
\[ \text{a) }\quad \operatorname{Int} A_{i_1}\cap \operatorname{Int} A_{i_2}=\varnothing \quad\text{for } i_1\ne i_2\ (i_1,i_2=1,2,\ldots,k-1); \]
\[ \text{b) }\quad \bigcup_{j=1}^{k-1} A_j\cup \bigcup_{i=k}^{s} O_i^1=X; \]
\[ \text{c) }\quad \operatorname{Fr}\operatorname{Int} A_j\subseteq \bigcup_{i=1}^{s}\operatorname{Fr} O_i^1 \quad (j=1,2,\ldots,k-1). \]

Construct \(A_k\):
\[ \operatorname{Int} A_k = O_k^1\setminus \left(\bigcup_{i=k+1}^{s}[O_i^1]\cup \bigcup_{j=1}^{k-1} A_j\right), \qquad A_k=[\operatorname{Int} A_k]. \]
Then, by the definition of the set \(A_k\), we obtain:
\[ \text{a) }\quad \bigcup_{j=1}^{k} A_j\cup \bigcup_{i=k+1}^{s} O_i^1=X. \]
Further,
\[ \operatorname{Fr}\operatorname{Int} A_k \subseteq \bigcup_{i=k}^{s}\operatorname{Fr} O_i^1 \cup \bigcup_{i=1}^{k-1}\operatorname{Fr}\operatorname{Int} A_j, \]
but
\[ \operatorname{Fr}\operatorname{Int} A_j\subseteq \bigcup_{i=1}^{s}\operatorname{Fr} O_i^1 \quad (j=1,2,\ldots,k-1), \]
therefore
\[ \text{b) }\quad \operatorname{Fr}\operatorname{Int} A_k \subseteq \bigcup_{i=1}^{s}\operatorname{Fr} O_i^1 \]
and
\[ \text{c) }\quad \operatorname{Int} A_k\cap \operatorname{Int} A_j=\varnothing \quad (j=1,2,\ldots,k-1). \]

The partition \(\alpha=\{A_1,A_2,\ldots,A_s\}\) is the desired one. Indeed, \(\alpha\) is inscribed in \(\omega\), and
\[ \operatorname{Ind}\operatorname{Fr}(\operatorname{Int} A_j) \leq \operatorname{Ind}\left(\bigcup_{i=1}^{s}\operatorname{Fr} O_i^1\right) \leq 0, \]
since \(\operatorname{Ind}\operatorname{Fr} O_i^1\leq 0\).

Proof of Theorem 3. Let \(P=X\times Y\), where \(X,Y\) are bicompacta and \(\operatorname{Ind}X\leq 1,\ \operatorname{Ind}Y\leq 1\). Let a closed set \(F\subset P\) and its neighborhood \(OF\) be arbitrary. One may take a cover
\(\Omega=\{\omega_1\times \omega_2\}\), whose elements are the sets
\(\{O_i\times G_j\}\) \((i=1,2,\ldots,n;\ j=1,2,\ldots,m)\), where
\(\omega_1=\{O_1,\ldots,O_n\}\) is an open cover of the bicompactum \(X\), and
\(\omega_2=\{G_1,\ldots,G_m\}\) is an open cover of the bicompactum \(Y\). In this case \(\Omega\) can be chosen so that the following conditions will be fulfilled:
\[ \text{1) }\quad \text{if } (O_i\times G_j)\cap F=\varnothing,\ \text{then } [(O_i\times G_j)]\cap F=\varnothing; \]
\[ \text{2) }\quad \left[\bigcup_{i=1}^{n}\bigcup_{j=1}^{m}(O_j\times G_j)\right]\subseteq OF, \quad \text{where } [O_i\times G_j]\cap F\ne\varnothing. \]

Now, using Lemma 1, construct such partitions
\[ \alpha=\{A_1,A_2,\ldots,A_n\} \quad\text{and}\quad \beta_1=\{B_1,B_2,\ldots,B_m\} \]
of the spaces \(X\) and \(Y\), respectively, that:
a) \(\alpha_1\) is inscribed in \(\omega_1\), \(\beta\) is inscribed in \(\omega_2\);
b) \(\operatorname{Ind}\operatorname{Fr}(\operatorname{Int} A_i)\leq 0\) and
\(\operatorname{Ind}\operatorname{Fr}(\operatorname{Int} B_j)\leq 0\)
\((i=1,2,\ldots,n;\ j=1,2,\ldots,m)\).

The partition
\[ \alpha=\{\alpha_1\times \alpha_2\}, \]
consisting of the sets
\[ A_i\times B_j \quad (i=1,2,\ldots,n;\ j=1,2,\ldots,m) \]
of the bicompactum \(P\), has the property that
\[ G=P\setminus \bigcup A_i\times B_j \]
(where \((A_i\times B_j)\cap F=\varnothing\)) is open in \(P\) and \(G\subseteq OF\). The boundary of \(G\) consists of closed subsets of sets of the form
\[ \operatorname{Fr} A_i\times B_j \quad\text{and}\quad A_i\times \operatorname{Fr} B_j \quad (i=1,2,\ldots,n;\ j=1,2,\ldots,m), \]
and moreover these sets are such that if two of them intersect, then they intersect in a zero-dimensional closed set. Therefore, from Theorem 2 it follows that
\[ \operatorname{Ind}\operatorname{Fr} G\leq 1. \]
Theorem 3 is proved.

Remark 1. Note that Lemma 1 is true for any normal space \(X\), provided only that \(\operatorname{Ind}X\leq 1\).

We now introduce two definitions.

Definition 1. We shall say that a closed set \(F\) is normally situated in a normal space \(X\), if for every closed set \(\Phi\subseteq F\) the subspace \(X\setminus \Phi\) is normal.

Example. Let \(S\) be the bicompactum of O. V. Lokutsievskii \((^2)\). In this bicompactum the set \(I\times \omega_1\) has the property that \(S\setminus (I\times \omega_1)\) is normal, but the set \(I\times \omega_1\) does not lie normally in the bicompactum \(S\). If, however, a closed set \(F\subseteq S\) is such that \(F\cap (I\times \omega_1)=\varnothing\), then \(F\) lies normally in \(S\).

Definition 2. A closed set \(F\) will be called normally Ind-lying in a normal space \(X\) if, for every closed set \(\Phi\subseteq F\), the subspace \(X\setminus F\) is normal and
\[ \operatorname{Ind}(X\setminus \Phi)\leq \operatorname{Ind}X. \]

Theorem 4. If a normal space \(P\) is the sum of two closed subsets \(X\) and \(Y\) in it such that: 1) \(\operatorname{Ind}X\leq n\), \(\operatorname{Ind}Y\leq n\); 2) \(F=X\cap Y\) is a set normally Ind-lying in \(X\) and in \(Y\); 3) \(\operatorname{Ind}F\leq n\), then \(\operatorname{Ind}P\leq n\).

Before proving Theorem 4, we prove Theorem 5, from which Theorem 4 will easily follow.

Theorem 5. Let \(P\) be a normal space. If in \(P\) there exists a closed set \(F\) such that \(\operatorname{Ind}F\leq n\), \(\operatorname{Ind}(P\setminus F)\leq n\), and \(F\) lies normally in \(P\), then \(\operatorname{Ind}P\leq n\).

Proof. We shall prove the theorem by induction on the number \(n\). For the sum of empty sets this is obvious. Suppose that Theorem 5 has already been proved for \(n-1\), i.e. if \(P\) is a normal space and in \(P\) there exists a closed set \(F\), normally lying in \(P\), such that \(\operatorname{Ind}F\leq n-1\) and \(\operatorname{Ind}(P\setminus F)\leq n-1\), then \(\operatorname{Ind}P\leq n-1\). We now prove Theorem 5 for \(n\).

Let \(\Phi\) be closed in \(P\), and let \(O\Phi\) be an arbitrary neighborhood of it.

a) If \(\Phi\cap F=\varnothing\), then there exists a neighborhood \(O_1\Phi\subseteq O\Phi\) such that
\[ \operatorname{Ind}\operatorname{Fr} O_1\Phi\leq n-1. \]

b) Let \(\Phi\cap F\neq \varnothing\). Denote by \(\Phi_1\) and \(O\Phi_1\) the sets
\[ \Phi_1=\Phi\cap F,\qquad O\Phi_1=O\Phi\cap F. \]
Since \(\operatorname{Ind}F\leq n\), choose a neighborhood \(O_1\Phi_1\subseteq O\Phi_1\) of the set \(\Phi_1\) such that
\[ \operatorname{Ind}\operatorname{Fr}_{F} O_1\Phi_1\leq n-1. \]
Put
\[ \Gamma=F\setminus [O_1\Phi_1],\qquad G_1=P\setminus \operatorname{Fr}_{F}O_1\Phi_1. \]
Then \(G_1\) is a normal space, \(\Gamma\) and \(O_1\Phi_1\) are closed in \(G_1\), and \(\Gamma\cap O_1\Phi_1=\varnothing\), \(\operatorname{Ind}(P\setminus F)\leq n\). Consider the sets
\[ \Phi_2=\Phi\cup O_1\Phi_1,\qquad \Phi'_2=\Gamma\cup (G_1\setminus O\Phi). \]
We shall have \(\Phi_2\cap \Phi'_2=\varnothing\), and \(\Phi_2,\Phi'_2\) are closed in \(G_1\). Now choose open sets \(G_2\) and \(G'_2\) in \(G_1\) such that
\[ \Phi_2\subseteq G_2,\qquad \Phi'_2\subseteq G'_2,\qquad [G_2]_{G_1}\cap [G'_2]_{G_1}=\varnothing. \]
Next consider the sets
\[ G_3=G_2\setminus F,\qquad G'_3=G'_2\setminus F. \]
We have
\[ [G_3]_{(P\setminus F)}\cap [G'_3]_{(P\setminus F)}=\varnothing. \]
Since \(\operatorname{Ind}(P\setminus F)\leq n\), choose an open set \(G_4\) in \(P\setminus F\) such that
\[ [G_3]_{(P\setminus F)}\subseteq G_4,\qquad [G_4]_{(P\setminus F)}\cap [G'_3]_{(P\setminus F)}=\varnothing \]
and
\[ \operatorname{Ind}\operatorname{Fr}_{(P\setminus F)}G_4\leq n-1. \]
The set
\[ G_5=G_4\cup O_1\Phi_1 \]
is open in \(P\) and
\[ \Phi\subseteq G_5\subseteq O\Phi. \]
Notice that
\[ \operatorname{Fr}_{P}G_5=\operatorname{Fr}_{(P\setminus F)}G_4\cup \operatorname{Fr}_{F}O_1\Phi_1, \]
where
\[ \operatorname{Ind}\operatorname{Fr}_{F}O_1\Phi_1\leq n-1, \]
\(\operatorname{Fr}_{F}O_1\Phi_1\) lies normally in \(\operatorname{Fr}_{P}G_5\), and
\[ \operatorname{Ind}\bigl(\operatorname{Fr}_{P}G_5\setminus \operatorname{Fr}_{F}O_1\Phi_1\bigr) = \operatorname{Ind}\operatorname{Fr}_{(P\setminus F)}G_4\leq n-1. \]
Thus, \(\operatorname{Ind}\operatorname{Fr}_{P}G_5\leq n-1\). Theorem 5 is proved.

Remark 2. The bicompactum of Lokutsievskii \((^2)\) \(S\) is represented as the sum of the closed set \(F=I\times \omega_1\) and the normal set \((S\setminus F)\), \(\operatorname{Ind}F\leq 1\), \(\operatorname{Ind}(S\setminus F)\leq 1\), while \(\operatorname{Ind}S=2\). But, as was said in Remark 1, the set \(F\) does not lie normally in \(S\). Thus, the concept of a normally lying set is essential.

We now prove Theorem 4. Let
\[ P=X\cup Y,\qquad F=X\cap Y. \]
Let a closed set \(\Phi\subseteq P\) and its neighborhood \(O\Phi\) be arbitrary.

a) If \(\Phi\cap F=\varnothing\), then there exists a neighborhood \(O_1\Phi\subseteq O\Phi\) such that
\[ \operatorname{Ind}\operatorname{Fr}O_1\Phi\leq n-1. \]

b) Let
\[ \Phi\cap F=\Phi_1\neq\varnothing. \]
Put
\[ O\Phi_1=F\cap O\Phi. \]
Choose \(O_1\Phi_1\subseteq O\Phi_1\) in such a way that
\[ \operatorname{Ind}\operatorname{Fr}_{F}O_1\Phi_1\leq n-1. \]
Let
\[ F_1=F\setminus [O_1\Phi_1]. \]
Then \(F_1\) and \(O_1\Phi_1\) are closed in
\[ G=P\setminus \operatorname{Fr}_{F}O_1\Phi_1 \]
and
\[ F_1\cap O_1\Phi_1=\varnothing. \]
Consider now the sets
\[ \Phi_2=\Phi\cup O_1\Phi_1,\qquad \Phi'_2=F_2\cup (G\setminus O\Phi). \]
The sets \(\Phi_2,\Phi'_2\) are closed in \(G\) and
\[ \Phi_2\cap \Phi'_2=\varnothing. \]
Te-

now take the sets \(X_1=X\cap \Phi_2,\ X_1'=X\cap \Phi_2'\) and \(Y_1=Y\cap \Phi_2,\ Y_1'=Y\cap \Phi_2',\ G_X=X\cap G,\ G_Y=Y\cap G\). We have
\(G_X=X\setminus \operatorname{bd}_F O_1\Phi_1,\ G_Y=Y\setminus \operatorname{bd}_F O_1\Phi_1\); consequently, \(G_X,\ G_Y\) are normal and \(\operatorname{Ind}G_X\le n,\ \operatorname{Ind}G_Y\le n\). Therefore we can take an open subset \(\Gamma_X\) (\(\Gamma_Y\)) in \(G_X\) (\(G_Y\)) such that
\(X_1\subset \Gamma_X\) (\(Y_1\subset \Gamma_Y\)), \([\Gamma_X]\cap X_1'=\varnothing\), \(([\Gamma_Y]\cap Y_1'=\varnothing)\), and
\(\operatorname{Ind}\operatorname{bd}_{G_X}\Gamma_X\le n-1\)
\((\operatorname{Ind}\operatorname{bd}_{G_Y}\Gamma_Y\le n-1)\). The set
\(\Gamma=\Gamma_X\cup \Gamma_Y\) is open in \(G\), and hence open also in \(P\), and
\[ \operatorname{bd}_P\Gamma = \operatorname{bd}_{G_X}\Gamma_X \cup \operatorname{bd}_{G_Y}\Gamma_Y \cup \operatorname{bd}_F O_1\Phi_1, \]
where
\[ \operatorname{bd}_{G_X}\Gamma_X\cup \operatorname{bd}_{G_Y}\Gamma_Y=\varnothing, \]
therefore
\[ \operatorname{Ind}(\operatorname{bd}_{G_X}\Gamma_X\cap \operatorname{bd}_{G_Y}\Gamma_Y)\le n-1 \]
and
\[ \operatorname{Ind}\operatorname{bd}_F O_1\Phi_1\le n-1. \]
Further, we see that \(\operatorname{bd}_F O_1\Phi_1\) lies normally in \(\operatorname{bd}_P\Gamma\) and
\[ \operatorname{bd}_F O_1\Phi_1\cap (\operatorname{bd}_{G_X}\Gamma_X\cup \operatorname{bd}_{G_Y}\Gamma_Y) =\varnothing; \]
consequently, by Theorem 5,
\[ \operatorname{Ind}\operatorname{bd}_P\Gamma\le n-1. \]
Theorem 4 is proved. From Theorems 4 and 5 it follows that

Theorem \(5'\) (Dowker \((^1)\)). Let \(P\) be a hereditarily normal space. If \(F\subset P\) is closed and \(\operatorname{Ind}F\le n,\ \operatorname{Ind}(P\setminus F)\le n\), then \(\operatorname{Ind}P\le n\).

Theorem \(4'\). Let \(P=X\cup Y\) be a hereditarily normal space; \(X,\ Y\) are closed in \(P\) and \(\operatorname{Ind}(X\cap Y)\le n,\ \operatorname{Ind}(X\setminus F)\le n,\ \operatorname{Ind}(Y\setminus F)\le n\), where \(F=X\cap Y\). Then \(\operatorname{Ind}P\le n\).

In conclusion I express my gratitude to my supervisor V. I. Ponomarev for his attention.

Moscow State University
named after M. V. Lomonosov

Received
7 XII 1966

CITED LITERATURE

\(^{1}\) C. H. Dowker, Quart. J. Math. Oxford, 4, 267 (1953).
\(^{2}\) O. V. Lokutsievskii, DAN, 67, No. 2 (1949).