Reports of the Academy of Sciences of the USSR

1. Volume 123, No. 1

MATHEMATICS

Yu. SMIRNOV

AN EXAMPLE OF A ZERO-DIMENSIONAL NORMAL SPACE HAVING INFINITE DIMENSION IN THE SENSE OF COVERINGS

(Presented by Academician P. S. Aleksandrov on 9 VI 1958)

Dowker (¹) constructed a remarkable normal space \(M\) with dimensions* \(\operatorname{ind} M = 0\) and \(\dim M = 1\). It is topologically embedded in the zero-dimensional (in all senses) bicompactum \(D^\tau\), thereby disproving P. S. Aleksandrov’s hypothesis on the monotonicity of the dimension \(\dim\) in the class of normal spaces.

The aim of this paper is the following generalization of Dowker’s construction, leading to the example indicated in the title (see item H).

A. Let \(T\) be the space of all ordinal numbers \(\alpha\) not exceeding \(\omega_1\), and \(W = T \setminus \omega_1\). For each \(\alpha < \omega_1\) we shall call the set \(W_\alpha = \delta\{\beta : \beta > \alpha\}\) an \(\alpha\)-tail. Let \(P\) be an arbitrary metric space with a countable base.

We shall call a set \(M\) of the product \(W \times P\) convergent (to \(P\)) if for every point \(p \in P\) there is a tail \(W_\alpha\) such that \(W_\alpha \times p \subseteq M\).

Theorem 1. Every convergent set \(M\) of the product \(W \times P\) is normal, countably paracompact**, and \(\dim M = \dim P\); if, moreover, \(\operatorname{ind} P = 0\), then \(\operatorname{dim} M = \dim P\) also in the infinite-dimensional cases***.

To prove this, a series of auxiliary lemmas is needed.

B. The intersection of a countable or finite number of closed cofinal \(\omega_1\)-sets \(A_i\) of the space \(W\) is nonempty****.

C. If the sets \(\Gamma_i\) are open in \(W\) and \(W = \bigcup_i \Gamma_i\), \(i = 1, 2, \ldots\), then at least one of them contains some \(\alpha\)-tail \(W_\alpha\).

* “Zero-dimensionality” is here to be understood only in the sense of the “small” inductive dimension \(\operatorname{ind}\) (induction is carried out over points). We shall also consider the “large” inductive dimension \(\operatorname{Ind}\) (induction is carried out over closed sets) and, mainly, the dimension \(\dim\), defined by means of finite open covers.

** A space \(R\) is countably paracompact and normal if into every countable or finite open cover \(\{\Gamma_i\}\) one can inscribe a closed cover \(\{\Phi_i\}\) such that \(\Phi_i \subseteq \Gamma_i\) (²).

*** From the point of view of essential mappings there are three types of infinite-dimensional spaces. We shall write \(\dim R < \mathfrak n\) (respectively, \(\dim R < \mathfrak N\)) if for every countable system \(\pi\) of pairs of closed sets \(A_i, B_i\), \(A_i \cap B_i = \varnothing\), there exist closed sets \(C_i\) separating the space \(R\) between \(A_i\) and \(B_i\) and such that \(\bigcap C_i = \varnothing\) for all \(i\) (respectively, for \(i \le N = N(\pi)\)). For countably paracompact (respectively, normal) spaces the assertion \(\dim R < \mathfrak n\) (respectively, \(\dim R < \mathfrak N\)) is equivalent to the following: for any sequence of continuous (real-valued) functions \(g_i\) and any number \(\varepsilon, \varepsilon > 0\), one can find continuous functions \(f_i\) such that \(|g_i - f_i| < \varepsilon\) and \(\bigcap f_i^{-1}(0) = \varnothing\) (respectively, for \(i \le N(\pi)\)), where \(\varnothing\) is the empty set.

**** By virtue of cofinality there is a sequence \(\{\alpha_i\}\), \(\alpha_i < \omega_1\), and a limit ordinal \(\alpha\) such that \(\alpha_{2^n+i} \in A_i\) for \(i = 0, \ldots, 2^{n+1} - 1\), and hence \(\alpha \in \bigcap_i A_i\).

D. If the sets \(\Gamma_i\) are open in \(W\times P\) and \(M\subseteq\bigcup_i \Gamma_i\), \(i=1,2,\ldots\), then for any point \(p\in P\) there exist a neighborhood \(Op\), a tail \(W_\beta\), and a set \(\Gamma_i\) such that \(W_\beta\times Op\subseteq\Gamma_i\).

Indeed, for the point \(p\) there is a tail \(W_\alpha\) such that \(W_\alpha\times p\subseteq M\). By C there exist a number \(\beta\geq\alpha\) and a set \(\Gamma_i\) such that \(W_\beta\times p\subseteq\Gamma_i\). For each \(\lambda\), \(\lambda\geq\beta\), take the real number \(\varepsilon_\lambda\)—the largest of all numbers \(\varepsilon>0\) such that \(\lambda\times O_\varepsilon p\subseteq\Gamma_i\). The number \(\delta=\inf_\lambda \varepsilon_\lambda\) is positive, since otherwise there would be order numbers \(\lambda_k\), converging to a number \(\lambda<\omega_1\), such that \(\varepsilon_{\lambda_k}\to0\), whence it would follow that \((\lambda,p)\notin\Gamma_i\) and \(\lambda\geq\beta\), contrary to the inclusion \(W_\beta\times p\subseteq\Gamma_i\). Thus, \(\lambda\times O_\delta p\subseteq\Gamma_i\) for every \(\lambda\geq\beta\), which was required to be proved.

E. If the sets \(\Gamma_i\) are open in \(W\times P\) and \(M\subseteq\bigcup_i \Gamma_i\), where \(i=1,2,\ldots\), then there exist an order number \(\beta\), \(\beta<\omega_1\), and open sets \(U_i\) of the space \(P\) such that \(P=\bigcup_i U_i\) and \(W_\beta\times U_i\subseteq\Gamma_i\) for each \(i\).

Indeed, by D, for any point \(p\in P\) there exist numbers \(\beta(p)<\omega_1\) and \(i(p)<\omega_0\) and a neighborhood \(Op\) such that \(W_{\beta(p)}\times Op\subseteq\Gamma_{i(p)}\). Choose a countable number of sets \(Op_j\) covering the space \(P\), with countable base. Let \(\beta=\sup\beta(p_j)\), and let
\[ U_i=\bigcup_{i(p_j)=i} Op_j \]
for each \(i\). This is what is needed.

F. Every continuous function \(f\), defined on \(M\), is finally constant, i.e. there is a number \(\beta\), \(\beta<\omega_1\), such that if \(\beta\leq\gamma<\gamma'<\omega_1\) and \((\gamma,p)\in M\), then \(f(\gamma,p)=f(\gamma',p)\).

Indeed, let \(p\in P\). Then there is a number \(\beta_p<\omega_1\) such that, if \(\gamma\geq\beta_p\), then \((\gamma,p)\in M\) and \(f(\gamma,p)=f(\beta_p,p)\) (see \((^3)\), p. 300). Take a countable dense set of points \(p_k\) from \(P\) and the number \(\beta=\sup\beta_k\). Then, if \(\gamma\geq\beta\), then \((\gamma,p_k)\in M\) and \(f(\gamma,p_k)=f(\beta,p_k)\). If \(p\ne p_k\), but \(\beta\leq\gamma<\gamma'<\omega_1\) and \((\gamma,p)\in M\), then let \(p_{k_j}\to p\). Then we have: \(f(\gamma',p)=\lim_j f(\gamma',p_{k_j})=\lim_j f(\gamma,p_{k_j})=f(\gamma,p)\), which was required to be proved.

Proof of the theorem. We shall use the definition of dimension \(\dim\) by means of “partitions”\(^*\). Let \(\dim P<m\), where \(m\in\{0,1,2,\ldots,\mathfrak w,\mathfrak W\}\). We shall prove that \(\dim P<m\). Let \(A_i, B_i\), \(A_i\cap B_i=\varnothing\), \(i=1,2,\ldots\), be closed subsets of \(M\). There exist sets \(\Gamma_A^i,\Gamma_B^i\), open in \(W\times P\), such that
\[ M\cap\Gamma_A^i=M\setminus A_i \quad\text{and}\quad M\cap\Gamma_B^i=M\setminus B_i. \]
Applying Lemma E to each pair \(\{\Gamma_A^i,\Gamma_B^i\}\) and taking a sufficiently large number \(\beta\), \(\beta<\omega_1\), we find that for the closed subsets \(A_i'=P\setminus U_A^i\), \(B_i'=P\setminus U_B^i\) of \(P\) the relations
\[ A_i\cap(W_\beta\times P)\subseteq W_\beta\times A_i', \qquad B_i\cap(W_\beta\times P)\subseteq W_\beta\times B_i' \]
hold, and \(A_i'\cap B_i'=\varnothing\). There are closed sets \(C_i'\), partitioning the space \(P\) between \(A_i'\) and \(B_i'\), the required number of which have empty intersection. The sets \(W_\beta\times C_i'\) partition \(W_\beta\times P\) and, in the required number, have empty intersection. The product \(\mathscr E\{\alpha:\alpha\leq\beta\}\times P\) has a countable base and, hence, \(\dim(\mathscr E\{\alpha:\alpha\leq\beta\}\times P)<m\)\(^ {**}\). Hence, in the finite case, and under the condition \(\operatorname{ind}P=0\), also in the infinite case, we obtain that
\[ \dim\bigl(M\setminus(W_\beta\times P)\bigr)=\operatorname{ind}P<m. \]
Since \(M\setminus(W_\beta\times P)\) is open and closed in \(M\), there exist sets \(C_i''\), closed in \(M\), complementing \(M\setminus(W_\beta\times P)\),

\(^*\) \(\dim R<n\), if for every system of \((n+1)\) pairs \(\{A_i,B_i\}\) of closed sets \(A_i,B_i\), \(A_i\cap B_i=\varnothing\), one can find closed sets \(C_i\), partitioning the space \(R\) between \(A_i\) and \(B_i\), such that \(\bigcap C_i=\varnothing\). The equivalence of this definition with the usual one for normal spaces was proved by P. S. Aleksandrov.

\(^ {**}\) This is also true in the infinite-dimensional case, but it obviously does not follow from this that \(\dim(M\cap(\mathscr E\{\alpha:\alpha\leq\beta\}\times P))<m\). For this it is sufficient to require, for \(\mathfrak w\), that \(\operatorname{ind}P<\omega_0\), and, for \(\mathfrak W\), that \(\operatorname{ind}P<\omega_2\).

separating it between \(A_i\cap M\setminus (W_\beta\times P)\) and \(B_i\cap M\setminus (W_\beta\times P)\) with empty intersection in the required number. Then the sets \(C_i'\cup (W_\beta\times C_i')\) separate \(M\) between \(A_i\) and \(B_i\) and, in the required number, give an empty intersection, as was to be proved. Normality and countable paracompactness are proved analogously.

Conversely, let \(A_i', B_i'\), \(A_i'\cap B_i'=\varnothing\), be closed sets in \(P\). If \(\dim M<m\), then there exist continuous functions \(f_i\), defined on \(M\), such that
\[ f_i\bigl(M\cap (W\times A_i')\bigr)=1,\quad f_i\bigl(M\cap (W\times B_i')\bigr)=-1 \quad\text{and}\quad \bigcap f_i^{-1}(0)=\varnothing \]
for the required number of functions. By virtue of F one may put \(f_i'(p)=f_i(\gamma_p,p)\), where \(\gamma_p\) is sufficiently large, and prove the continuity of the functions \(f_i'\). We have
\[ f_i'(A_i')=1,\quad f_i'(B_i')=-1 \quad\text{and}\quad \bigcap f_i'^{-1}(0)=\varnothing \]
for the required number of functions \(f_i'\), i.e. \(\dim P<m\), as was to be proved.

Theorem 2. For every metric space \(P\) with a countable base, in the product \(W\times P\) there exists a zero-dimensional (i.e. \(\operatorname{ind} M=0\)) convergent set \(M\) (of the same dimension \(\dim\) as \(P\)).

G. Every metric space with a countable base can be represented as the sum of zero-dimensional sets forming a nondecreasing transfinite sequence of type \(\omega_1\). *

Indeed, for the interval \(E^1\) such a representation was obtained by Dowker **(1). Let
\[ E^1=\bigcup_\alpha P_\alpha^1,\quad \alpha<\omega_1,\quad P_\alpha^1\subset P_{\alpha'}^1\quad\text{for}\quad \alpha<\alpha'. \]
For the Hilbert cube \(E^\infty=(E^1)^{\aleph_0}\) an analogous representation
\[ E^\infty=\bigcup_\alpha P_\alpha^\infty \]
is made up of powers *** of the sets \(P_\alpha^1\):
\[ P_\alpha^\infty=(P_\alpha^1)^{\aleph_0}. \]
For an arbitrary space \(P\) with a countable base, the required representation
\[ P=\bigcup_\alpha P_\alpha \]
is obtained by intersection: if \(P\subset E^\infty\), then \(P\cap P_\alpha^\infty=P_\alpha\).

Proof of the theorem. Let
\[ P=\bigcup_\alpha P_\alpha \]
be the representation obtained in G. The set
\[ M=\bigcup_\alpha (W_\alpha\times P_\alpha) \]
is convergent and \(\operatorname{ind} M=0\), as was to be proved.

H. The desired example is obtained by taking the representation of the Hilbert cube \(E^\infty\) constructed in G and the corresponding convergent set \(M\) of the product \(W\times E^\infty\).

Moscow State University
named after M. V. Lomonosov

Received
6 V 1958

REFERENCES

1. C. H. Dowker, Canad. J. Math., 3, 219 (1951).
2. C. H. Dowker, Quart. J. Math., 6, No. 22, 101 (1955).
3. Yu. Smirnov, Matem. sborn., 38, No. 3, 283 (1956).

* Under the assumption of the continuum hypothesis this is obvious.

** Let \(Q_0\) be the set of rational points of the interval \(E^1=[0;1]\), and let \(Q_\alpha\) be the congruence classes modulo \(Q_\alpha\), numbered by ordinal numbers \(\alpha\), \(0\leq \alpha<\omega(c)\). The sets
\[ P_\alpha^1=\bigcup_{\gamma\geq \omega_1} Q_\gamma\cup \bigcup_{\beta<\alpha} Q_\beta \]
are zero-dimensional for \(\alpha<\omega_1\), and, moreover,
\[ E^1=\bigcup P_\alpha^1. \]

*** That is, the Tychonoff product of the spaces \(P_\alpha^1\) with themselves, taken the required number of times. This construction was suggested to me by S. Proizvolov. Obviously, \(\operatorname{ind} P_\alpha^\infty=0\); if \(\alpha<\alpha'\), then \(P_\alpha^\infty\subset P_{\alpha'}^\infty\); if \(\{x_i\}\in E^\infty\), then, taking numbers \(\alpha_i\) such that \(x_i\in P_{\alpha_i}^1\), we find that \(\{x_i\}\in P_\alpha^\infty\) for \(\alpha=\sup \alpha_i\).