MATHEMATICS

E. G. SKLYARENKO

TWO THEOREMS ON INFINITE-DIMENSIONAL SPACES

(Presented by Academician P. S. Aleksandrov, 23 XI 1961)

1. A normal space \(X\) is called weakly infinite-dimensional if, for every countable system of pairs of closed sets \((A_i, B_i)\), \(A_i \cap B_i = \varnothing\), one can find a system of closed sets \(C_i\) (the set \(C_i\) will be called a partition between \(A_i\) and \(B_i\)) such that each \(C_i\) separates the corresponding sets \(A_i\) and \(B_i\) and \(\bigcap_i C_i = \varnothing\); otherwise the space \(X\) is called strongly infinite-dimensional.*

Theorem 1. Under every continuous mapping \(f\) of a weakly infinite-dimensional bicompactum \(X\) onto a strongly infinite-dimensional bicompactum \(Y\), there is in \(Y\) a point \(y\) whose complete inverse image \(f^{-1}y\) has cardinality not less than the cardinality of the continuum.

In other words, if, under a continuous mapping \(f\) of a weakly infinite-dimensional bicompactum \(X\) onto a bicompactum \(Y\), the complete inverse images of all points of \(Y\) have cardinality less than the continuum, then the bicompactum \(Y\) is also weakly infinite-dimensional.

An analogous theorem for another class of infinite-dimensional compacta was proved in \((^{6})\), namely, instead of weakly (and, respectively, strongly) infinite-dimensional spaces, countable-dimensional (and, respectively, non-countable-dimensional) compacta were considered there. As B. T. Levshenko showed \((^{3})\), every countable-dimensional space is weakly infinite-dimensional. The question of whether the converse is true, i.e. whether the classes of weakly infinite-dimensional and countable-dimensional compacta coincide or not, was posed by P. S. Aleksandrov (see \((^{1})\), p. 54, or \((^{2})\), p. 14) and remains open to this day. In the case that these classes coincide, Theorem 1 will turn out to be a generalization of the result from \((^{6})\) cited above to bicompact spaces.*

Proof of Theorem 1. It is enough to prove the following lemma.

Lemma 1. Under the conditions of Theorem 1, there exist in the space \(X\) two disjoint closed sets \(X_0\) and \(X_1\) such that \(fX_0 = fX_1 = Y_1\), and the set \(Y_1\) is strongly infinite-dimensional.

Indeed, suppose that Lemma 1 has been proved. Then, by induction, one can construct a decreasing sequence of strongly infinite-dimensional closed subsets \(Y_n\) in \(Y\), and in \(X\) a system of closed subsets \(X_{i_1\ldots i_n}\) \((i_k = 0, 1)\) such that \(X_{i_1\ldots i_n} \cap X_{j_1\ldots j_n} = \varnothing\), if \((i_1 \ldots i_n) \ne (j_1 \ldots j_n)\), \(X_{i_1\ldots i_n} \subset X_{i_1\ldots i_{n-1}}\), and \(fX_{i_1\ldots i_n} = Y_n\). If this is done, then, as is easy to verify, any point \(y \in \bigcap_n Y_n\) will have a complete

* This definition was proposed by P. S. Aleksandrov (see \((^{1})\), p. 54, or \((^{2})\), p. 14). Yu. M. Smirnov indicated another variant of the definition of weak (and, respectively, strong) infinite-dimensionality (see \((^{3,5})\)), namely: a space is called weakly infinite-dimensional in the sense of Yu. M. Smirnov if, for every countable system of pairs of closed sets \((A_i, B_i)\), \(A_i \cap B_i = \varnothing\), one can find such a finite system of partitions \(C_i\), \(i \le k\) (where \(k\) depends on the system of pairs under consideration), between \(A_i\) and \(B_i\), that \(\bigcap_{i\le k} C_i = \varnothing\). For compact spaces these two definitions are equivalent.

** A space is called countable-dimensional if it is the sum of a countable number of zero-dimensional \((\dim R_i = 0)\) sets \(R_i\). W. Hurewicz proved that a separable space possessing a complete metric is countable-dimensional if and only if it has transfinite dimension \((^{2})\).

*** A generalization of another kind (to metric spaces) was obtained by Yu. M. Smirnov in \((^{7})\).

preimage of cardinality not less than the continuum. Thus, suppose that the sets \(Y_k\) and \(X_{i_1\ldots i_k}\), \(k=1,\ldots,n-1\), have already been constructed. Applying Lemma 1 to the mapping \(f:X_{0\cdots0}\to Y_{n-1}\), we find in \(X_{0\cdots0}\) two disjoint closed subsets \(X'_{0\cdots00}\) and \(X'_{0\cdots01}\) such that \(fX'_{0\cdots00}=fX'_{0\cdots01}=Y'_n\subset Y_{n-1}\), and the set \(Y'_n\) is strongly infinite-dimensional. Then we apply Lemma 1 to the sets \(Y'_n\) and \(X_{0\cdots1}\cap f^{-1}Y'_n\); we obtain the sets \(Y''_n\), \(X'_{0\cdots10}\), and \(X'_{0\cdots11}\). Repeating this procedure \(2^{n-1}\) times, we finally obtain a strongly infinite-dimensional closed subset \(Y_n\subset Y_{n-1}\) and sets \(X'_{i_1\ldots i_n}\) such that \(X'_{i_1\ldots i_{n-1}i_n}\subset X_{i_1\ldots i_{n-1}}\). Put then
\[ X_{i_1\ldots i_n}=f^{-1}Y_n\cap X'_{i_1\ldots i_n}, \]
and the systems of subsets in \(X\) and \(Y\) that we need have been constructed.

Proof of Lemma 1. Since the bicompactum \(Y\) is strongly infinite-dimensional, in \(Y\) there exists a countable system of pairs of closed sets \((A_i,B_i)\), \(A_i\cap B_i=\varnothing\), which cannot be separated by partitions with empty intersection.* Let \((A'_i,B'_i)\) be the system of pairs of closed sets in \(X\) which are the complete preimages of the sets \(A_i,B_i\). Since the bicompactum \(X\) is weakly infinite-dimensional, for some finite subsystem \((A'_i,B'_i)\), \(i=1,\ldots,n\), there will be partitions \(C'_i\) in \(X\) with empty intersection. Let \(M,N\) be closed sets in \(X\) such that \(A'_1\subset M\), \(B'_1\subset N\), \(M\cap N=C'_1\), \(M\cup N=X\). Let, further, \(C=fC'_1\) and \(D=fM\cap fN\). The set \(D\) is a partition in \(Y\) between \(A_1\) and \(B_1\). Since in \(Y\) the system of pairs \((A_i,B_i)\), \(i=1,2,\ldots\), cannot be separated by partitions with empty intersection, in the bicompactum \(D\) the following system is inseparable by partitions with empty intersection:
\[ (A_i\cap D,\; B_i\cap D),\quad i=2,3,\ldots \]
(see the proof of Lemma 5 from \((^5)\)). Therefore the bicompactum \(D\) is strongly infinite-dimensional.

The proof of the lemma is by induction on the number \(n\). If \(n=1\), then \(C'_1=M\cap N=\varnothing\), and we may put
\[ Y_1=D,\qquad X_0=f^{-1}D\cap M,\qquad X_1=f^{-1}D\cap N. \]
Suppose now that the lemma is true if the number of “separated” pairs in \(X\) is less than \(n\). First case: the pairs
\[ (A_i\cap C,\; B_i\cap C),\quad i=2,3,\ldots, \]
are inseparable on the set \(C\) by partitions with empty intersection. Hence the bicompactum \(C\) is strongly infinite-dimensional. Consider then the mapping \(f\) on the set \(C'_1\). The complete preimages of the pairs \((A_i\cap C,\; B_i\cap C)\), \(i\ge2\), in \(C'_1\) are the pairs
\[ (A'_i\cap C'_1,\; B'_i\cap C'_1). \]
The sets \(C'_i\cap C'_1\), \(i=2,\ldots,n\), are partitions for the first \(n-1\) pairs of this system, and moreover
\[ \bigcap_{i=2}^{n}(C'_i\cap C'_1)=\bigcap_{i=1}^{n}C'_i=\varnothing. \]
Therefore, by the induction hypothesis, in \(C\) and \(C'_1\) there will be found sets satisfying the conditions of the lemma. Consider the remaining case: in \(C\) there exist partitions \(E_i\), \(i=2,\ldots,m\), between the sets \(A_i\cap C\) and \(B_i\cap C\) such that
\[ \bigcap_{i=2}^{m}E_i=\varnothing. \]
We may suppose that each \(E_i\) has type \(G_\delta\). Then in \(D\) there exist partitions \(F_i\) between the sets \(A_i\cap D\) and \(B_i\cap D\), \(i=2,\ldots,m\), such that \(F_i\cap C=E_i\). Since the pairs \((A_i\cap D,\;B_i\cap D)\), \(i=2,3,\ldots\), cannot be separated in \(D\) by partitions with empty intersection, the set
\[ F=\bigcap_{i=2}^{m}F_i \]
is strongly infinite-dimensional (see again the proof of Lemma 5 from \((^5)\)). Moreover, \(F\cap C=\varnothing\). Therefore we may put
\[ Y_1=F,\qquad X_0=f^{-1}F\cap M,\qquad X_1=f^{-1}F\cap N. \]
This completes the proof of the lemma, and with it also the proof of Theorem 1.**

* We shall constantly use this convenient expression (or its modifications), which in the present case means that there are no partitions \(C_i\) between \(A_i\) and \(B_i\) such that
\[ \bigcap_i C_i=\varnothing. \]

** The proof of Lemma 1 given above remains valid if \(X\) and \(Y\) are normal spaces, respectively weakly and strongly infinite-dimensional in the sense of Yu. M. Smirnov, and the mapping \(f\) is closed.

Remark 1. Theorem 1 can be extended to the noncompact case as follows:

Theorem 1′. For every closed continuous mapping \(f\) of a weakly infinite-dimensional (in the sense of Yu. M. Smirnov) complete metric space with a countable base \(X\) onto a strongly infinite-dimensional (in the sense of P. S. Aleksandrov) normal countably paracompact space \(Y\), there is a point \(y\) in \(Y\) whose full preimage has cardinality not less than the cardinality of the continuum.*

For the proof of the theorem the following modification of Lemma 1 is used:

Lemma 1′. Let \(f\) be a closed mapping of a weakly infinite-dimensional (in the sense of Yu. M. Smirnov) normal space \(X\) onto a strongly infinite-dimensional (in the sense of P. S. Aleksandrov) countably paracompact space \(Y\), and let \(\omega\) be a countable closed covering of the space \(X\). Then in \(X\) there are two disjoint closed sets \(X_0\) and \(X_1\), contained in some elements of the covering \(\omega\), such that \(fX_0=fX_1=Y_1\), and the closed set \(Y_1\) in \(Y\) is strongly infinite-dimensional.

First, repeating with minor changes the arguments from the proof of Lemma 1, we construct in \(X\) disjoint closed sets \(X'_0\) and \(X'_1\) such that the set \(Y'_0=fX'_0=fX'_1\) is strongly infinite-dimensional (in the sense of P. S. Aleksandrov). Let

\[ X'_0=\bigcup_{i=1}^{\infty} X_0^i, \]

where each \(X_0^i\) is contained in some element of the covering \(\omega\). Then

\[ Y'_1=\bigcup_i fX_0^i, \]

and since \(Y'_1\) is strongly infinite-dimensional, for some \(i\) the set \(Y''_1=fX_0^i\) is strongly infinite-dimensional. Repeating the analogous arguments for \(Y''_1\) and \(X'_1\cap f^{-1}Y''_1\), we construct the required sets \(Y_1, X_0, X_1\).

The proof of Theorem 1′ proceeds analogously to the proof of Theorem 1, with the sole change that the sets \(X_{i_1\ldots i_n}\) must have diameter less than \(1/n\).

Remark 2. With the help of arguments analogous to those used above, one can give a simple proof of the following theorem of K. Morita (see \((^4)\)):

Let \(f\) be a closed \((m+1)\)-fold mapping of a normal space \(X\) onto a normal space \(Y\). Then

\[ \dim Y \leq \operatorname{Ind} X + m. \]

Let \(\operatorname{Ind} X\leq n\) and \(\dim Y\geq n+m\); we shall show that then the multiplicity of the mapping \(f\) is not less than \(m+1\). We prove this by induction on \(n\) and \(m\). If \(\operatorname{Ind} X=-1\) or if \(m=0\), then the assertion is true. Suppose that the assertion is true if \(\operatorname{Ind} X\leq n-1\), or if \(\operatorname{Ind} X\leq n\) and \(\dim Y\geq n+m-1\). Let \(\operatorname{Ind} X\leq n\) and \(\dim Y\geq n+m\). In the space \(Y\) there are \(n+m\) pairs of closed sets \((A_i,B_i)\), \(A_i\cap B_i=\varnothing\), which cannot be separated by partitions with empty intersection. Further we use the notation from the proof of Lemma 1; moreover, since \(\operatorname{Ind} X\leq n\), one may assume that \(\operatorname{Ind} C'_1\leq n-1\). If the pairs \((A_i\cap C, B_i\cap C)\), \(i=2,\ldots,n+m\), cannot be separated in \(C\) by partitions with empty intersection, then, according to the induction hypothesis (using the fact that \(\operatorname{Ind} C'_1\leq n-1\)), the mapping \(f\) already on the set \(C'_1\) has multiplicity \(\geq m+1\). If, however, these pairs can be separated in \(C\) by partitions with empty intersection, then the pairs \((A_i\cap D, B_i\cap D)\) can be separated in \(D\) by partitions \(F_i\), whose intersection \(F\) does not meet \(C\). The set \(F\) is nonempty, since the pairs \((A_i\cap D, B_i\cap D)\) cannot be separated in \(D\) by partitions with empty intersection (cf. the corresponding place in the proof—

\[ \text{*} \]

This is important: there exists a strongly infinite-dimensional space \(Y\) in the sense of Yu. M. Smirnov (the discrete sum of a countable number of cubes of increasing dimension), onto which the zero-dimensional space \(X\) (the sum of a countable number of Cantor sets) can be mapped so that the mapping is closed and the full preimage of each point is finite.

of Lemma 1). Let \(U\) be a closed neighborhood of the set \(F\) in \(D\), not intersecting \(C\). Then in the set \(D\) there exist pairs of disjoint closed sets \((K_i,L_i)\), \(i=2,\ldots,n+m\), such that
\(A_i\cap D\subset K_i\), \(B_i\cap D\subset L_i\), and \(\bigcap_{i=2}^{n+m}\{D\setminus(K_i\cap L_i)\}\subset U\). Since the intersection of any partitions for the pairs \((K_i,L_i)\) lies in \(U\), and since these partitions are at the same time also partitions for the pairs \((A_i\cap D,B_i\cap D)\), the \(n+m-1\) pairs of closed sets \((K_i\cap U,L_i\cap U)\) cannot be separated in \(U\) by partitions with empty intersection; consequently, \(\dim U\ge n+m-1\). On the other hand, since \(U\cap C=\varnothing\), in \(X\) there are two disjoint closed sets \(f^{-1}U\cap M\) and \(f^{-1}U\cap N\), each of which is mapped onto \(U\). Since \(\dim U\ge n+m-1\), and \(\operatorname{Ind}(f^{-1}U\cap M)\le n\), by the induction hypothesis there is in \(U\) a point having in \(f^{-1}U\cap M\) at least \(m\) preimages; moreover, this point has at least one preimage in \(f^{-1}U\cap N\). Consequently it has at least \(m+1\) preimages in \(X\), as was required to prove.

1. B. T. Levshenko proved \({}^{3}\) the following proposition:

A normal space \(X\) is strongly infinite-dimensional in the sense of Yu. M. Smirnov if and only if there exists a mapping \(f\) of the space \(X\) into the Hilbert parallelepiped \(Q\) such that the following condition is satisfied:

B. For every finite-dimensional face \(F\) in \(Q\) and projection \(\pi: Q\to F\), the mapping \(\pi f\) of the space \(X\) into the cube \(F\) is essential.

Theorem 2. A bicompactum \(X\) is strongly infinite-dimensional if and only if there exists a mapping \(f\) of the bicompactum \(X\) into the Hilbert parallelepiped \(Q\) such that the following condition is satisfied:

G. For every finite-dimensional face \(F\) in \(Q\), the mapping \(f\) of the set \(f^{-1}F\) into the cube \(F\) is essential.

Theorem 2 follows, obviously, from the following proposition:

Theorem \(2'\). For every mapping \(f\) of a bicompactum \(X\) into \(Q\), conditions B and G are equivalent.

Proof. Obviously, condition B follows from condition G. We shall show that the converse is true. Let the mapping \(f=\{f_1,\ldots,f_n,\ldots\}\) satisfy condition B. As B. T. Levshenko showed \({}^{3}\), this is equivalent to saying that the system of pairs \((A_i,B_i)\), \(A_i=\{x:f_ix=0\}\), \(B_i=\{x:f_ix=1\}\), cannot be separated in \(X\) by partitions with empty intersection. Suppose that, onto some finite-dimensional face \(F=\{q\in Q:q_i=0,\ i>n\}\), its full preimage \(\Phi=f^{-1}F\) is mapped inessentially. Then on the set \(\Phi\) there are functions \(g_i'\), \(i=1,\ldots,n\), such that they map \(\Phi\) into the boundary \(S\) of the cube \(F\) and coincide with the functions \(f_i\) on the set \(f^{-1}S\). Construct on the bicompactum \(X\) functions \(g_i\), \(i=1,\ldots,n\), whose values lie between zero and one, such that \(g_i=g_i'\) if \(x\in\Phi\); \(g_i=0\) if \(x\in A_i\), and \(g_i=1\) if \(x\in B_i\). Put \(g_i=f_i\) if \(i\ge n+1\). Consider the mapping \(g=\{g_1,\ldots,g_n,\ldots\}\) of the bicompactum \(X\) into \(Q\), and the pairs \((A_i',B_i')\), \(A_i'=\{x:g_ix=0\}\), \(B_i'=\{x:g_ix=1\}\), of closed sets in \(X\). For every \(i\) we have \(A_i\subset A_i'\), \(B_i\subset B_i'\), and therefore the system of pairs \((A_i',B_i')\) cannot be separated in \(X\) by partitions with empty intersection. Since \(X\) is a bicompactum, it follows from this that \(g\) must map \(X\) onto the whole Hilbert parallelepiped \(Q\). On the other hand, by the construction of \(g\), it does not cover the point \((1/2,1/2,\ldots,1/2,0,0,\ldots)\). The theorem is proved.

Moscow State University
named after M. V. Lomonosov

Received
21 XI 1961

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