Reports of the Academy of Sciences of the USSR

1. Volume 152, No. 4

MATHEMATICS

B. Efimov

METRIZABILITY AND THE \(\Sigma\)-PRODUCT OF BICOMPACTS

(Presented by Academician P. S. Aleksandrov on 17 IV 1963)

§ 1. \(\Sigma_{\mathfrak m}\)-products of bicompacts. Let \(X=\prod_{\alpha\in\Theta} X_\alpha\) be the product of topological spaces \(X_\alpha\), with \(\operatorname{card}\Theta\ge \aleph_1\). Let \(p=(p_\alpha)\in X\). Following Corson \((^1)\), by the \(\Sigma\)-product of the spaces \(X_\alpha\) with base point \(p\) we shall mean the subset \(\Sigma^p\subset X\) consisting of those points \(x=(x_\alpha)\in X\) for which \(x_\alpha\ne p_\alpha\) for at most a countable set of indices \(A_x\subset\Theta\). Let \(\mathfrak m\ge \aleph_0\). Then by the \(\Sigma_{\mathfrak m}\)-product of the spaces \(X_\alpha\) with base point \(p=(p_\alpha)\) we shall mean the subset \(\Sigma^p_{\mathfrak m}\subset X\) consisting of those points \(x=(x_\alpha)\in X\) for which \(x_\alpha\ne p_\alpha\) for at most a set of indices of cardinality \(\le \mathfrak m\).

Theorem 1. The weight of a bicompactum that is a continuous image of the \(\Sigma_{\mathfrak m}\)-product of bicompacts \(X_\alpha\) of weight \(\mathfrak m_\alpha\ge 2\) does not exceed
\[ \max\left(\mathfrak m,\ \sup_\alpha \mathfrak m_\alpha\right). \]

In particular, if the weight of \(X_\alpha\) is \(\le \aleph_0\) for all \(\alpha\in\Theta\) and \(\mathfrak m=\aleph_0\), then as a corollary we obtain a result partially generalizing Corson’s result \((^1)\), who proved that a metric space that is a continuous image of the \(\Sigma\)-product of complete separable metric spaces is separable.

Corollary. A bicompactum that is a continuous image of the \(\Sigma\)-product of compacta is metrizable.

With the aid of Theorem 1 the following basic result is proved.

Theorem 2. The weight of a bicompactum \(R\) that is a continuous image of the bicompactum
\[ X=\prod_{\alpha\in\Theta} X_\alpha, \]
where \(X_\alpha\) is a bicompactum of weight \(\mathfrak m_\alpha\ge \aleph_0\), does not exceed
\[ \max\left[\sup_\alpha \mathfrak m_\alpha,\ \sup_{x\in R}\chi(x,R)\right]. \]

In particular, if \(X=D^\tau\), \(\tau\ge \aleph_1\) (\(D^\tau\) is the generalized Cantor discontinuum of weight \(\tau\)), then we obtain the well-known theorem that the weight of a dyadic bicompactum is equal to \(\sup_{x\in R}\chi(x,R)\).*

Lemma 1. Let \(R\) be a bicompactum of weight \(\ge \tau\). Then the following two assertions are valid: 1) \(R\) contains a bicompactum \(R'\) of density \(\le \tau\) and weight \(\ge \tau\); 2) \(R\) maps onto a bicompactum \(R''\) of weight \(\tau\).**

Proof. We note that 2) implies 1). Indeed, let \(R''=f(R)\), with the weight of \(R''\) equal to \(\tau\). Let \(M\) be dense in \(R''\) and \(\operatorname{card} M\le \tau\). For each \(x\in M\) choose a point \(y\in f^{-1}(x)\). Denote the set of all chosen points by \(L\). It is easy to see that \(\operatorname{card} L\le \tau\) and that \(f[L]=[M]_{R''}=R''\), and therefore the weight of \([L]_R\ge \tau\). Thus, \([L]=R'\) is the required one.

We now prove 2). Let \(C(R)\) be the ring of all continuous real functions on \(R\) with metric
\[ \rho(g,g')=\sup_{x\in R}|g(x)-g'(x)|. \]
We note,

* \(\chi(x,R)\) denotes the local character of the point \(x\) relative to \(R\).
** The density of \(R\) is the least cardinality of an everywhere dense set.

that the weight of \(C(R)=\) weight \(R\ge \tau\) \((^2)\). Let \(\mathfrak B=\{f_\alpha\}\), \(f_\alpha\in C(R)\), be a maximal family of functions such that for any \(f_\alpha, f_\beta\in \mathfrak B\), \(\rho(f_\alpha,f_\beta)\ge 1\). We shall show that \(\operatorname{card}\mathfrak B\ge \tau\). Suppose that \(\operatorname{card}\mathfrak B\le \tau'<\tau\). Add to this family the function \(f\equiv 1\) and consider the subring \(C'(R)\) of the ring \(C(R)\) formed from all possible polynomials with rational coefficients in the family \(\mathfrak B\). Note that the family \(C'(R)\) does not separate all points of \(R\), since otherwise, by M. Stone’s theorem \((^3)\), one would have \([C'(R)]=C(R)\), but \(\operatorname{card} C'(R)=\tau'\) and, consequently, the weight of \(C(R)=\tau'<\tau\), for for a metric space the weight coincides with the density. Thus there exist two points \(x,y\in R\) such that for every function \(f\in C'(R)\), \(f(x)=f(y)\). Now consider a function \(g\in C(R)\) such that \(g(x)=1\) and \(g(y)=-1\). It is easy to see that for every \(f_\alpha\in\mathfrak B\), \(\rho(f_\alpha,g)\ge 1\), which contradicts the maximality of the family \(\mathfrak B\). Thus \(\operatorname{card}\mathfrak B\ge \tau\). Without loss of generality we assume that \(\operatorname{card}\mathfrak B=\tau\). On the other hand, to each function \(f_\alpha\in C(R)\) we associate the interval \(I_\alpha=f(R)\) and construct, by the method of A. N. Tikhonov \((^4)\), a topological embedding of \(R\) in \(I^\tau=\prod_\alpha I_\alpha\). Now consider the face \(I^\tau\) of the bicompactum \(I^\tau\)

\[ \prod_\alpha I_\alpha, \]

which is \(\prod_\alpha I_\alpha\), if \(I_\alpha\) corresponds to \(f_\alpha\in\mathfrak B\). Let \(\mathfrak F\) be the natural projection of \(I^\tau\) onto \(I^\tau\). We shall show that \(\mathfrak F(R)=R''\) is the required bicompactum. Indeed, \(R''\subset I^\tau\); consequently, the weight of \(R''\le \tau\), and therefore the density of \(R''\le \tau\). However, in the ring \(C(R'')\) there exists a family of functions \(\mathfrak B\) of cardinality \(\tau\), therefore the weight of \(C(R'')\ge \tau\), whence the weight of \(R''\ge \tau\). The lemma is proved.

Proof of Theorem 1. Suppose the contrary. Let \(R=f(\Sigma_{\mathfrak m}^{p})\), with weight \(R>\mathfrak l=\max(\mathfrak m,\sup_\alpha \mathfrak m_\alpha)\). Denote by \(\mathfrak n\) the least cardinal number greater than \(\mathfrak l\). Then the weight of \(R\ge \mathfrak n\). By Glicksberg’s theorem \((^5)\), the Stone–Čech compactification of \(\Sigma^p\) is \(X\). Since \(\Sigma_{\mathfrak m}^{p}\subset \Sigma^p\subset X\), we have \(\beta\Sigma_{\mathfrak m}^{p}=\beta\Sigma^{p}=X\) (see \((^6)\), p. 89). Denote by \(\varphi\) the Stone extension of the mapping \(f\) of the bicompactum \(X\) onto \(R\). Using Lemma 1, consider in \(R\) a bicompactum \(R'\) of density \(\le \mathfrak n\) and weight \(\ge \mathfrak n\). Let \(M\) be a dense subset of \(R'\) of cardinality \(\le \mathfrak n\). For each point \(x_\nu\in M\), choose a point \(y^\nu\in f^{-1}(x_\nu)\in \Sigma_{\mathfrak m}^{p}\). Let \(A_\nu\) be the set of all indices for which \(y_\alpha^\nu\ne p_\alpha\), if \(\alpha\in A_\nu\). Since \(y^\nu\in \Sigma_{\mathfrak m}^{p}\), \(\operatorname{card} A_\nu\le \mathfrak m\). Put \(A=\bigcup_\nu A_\nu\); then \(\operatorname{card} A\le \mathfrak n\), since \(\mathfrak m<\mathfrak n\). Consider

\[ X_0=\prod_{\alpha\in A} X_\alpha . \]

This set may be regarded as a subspace of \(X\) containing all the points \(y^\nu\), by putting, for all \(y=(y_\alpha)\in X_0\), \(y_\alpha=p_\alpha\), if \(\alpha\in \Theta\setminus A\). Further, since \(2^{\mathfrak l}\ge \mathfrak n\) and the density of \(X_\alpha\le \mathfrak m_\alpha\le \mathfrak l\) for all \(\alpha\in A\), the density of \(X_0\le \mathfrak l\) \((^7)\). Thus the bicompactum \(\Phi=\varphi(X_0)\), first, has density \(\le \mathfrak l\), and, second, contains \(R'\), hence has weight \(\ge \mathfrak n\). We shall show that this cannot be. Let \(L\) be a dense subset of \(\Phi\) of cardinality \(\le \mathfrak l\). For each \(a_\nu\in L\), choose a point \(b^\nu\in f^{-1}(a_\nu)\in \Sigma_{\mathfrak m}^{p}\). Let \(B_\nu\) be the set of all indices for which \(b_\alpha^\nu\ne p_\alpha\), if \(\alpha\in B_\nu\). Since \(b^\nu\in \Sigma_{\mathfrak m}\), \(\operatorname{card} B_\nu\le \mathfrak m\). Put \(B=\bigcup_\nu B_\nu\); then \(\operatorname{card} B\le \mathfrak l\), since \(\mathfrak m\le \mathfrak l\). Consider

\[ Y_0=\prod_{\alpha\in B} X_\alpha . \]

As before, this set may be regarded as a subspace of \(X\) containing all the points \(b^\nu\), by putting, for all \(y=(y_\alpha)\in Y_0\), \(y_\alpha=p_\alpha\), if \(\alpha\in \Theta\setminus B\). Note that \(Y_0\) is a bicompactum of weight \(\le \mathfrak l\), since the weight of \(X_\alpha\le \mathfrak m_\alpha\le \mathfrak l\) and \(\operatorname{card} B\le \mathfrak l\). Therefore \(F=[\{b^\nu\}]_X\subset Y_0\) and the weight of \(F\le \mathfrak l\). On the other hand,

\(\varphi F=[L]=\Phi\), consequently, \(\operatorname{weight}\Phi \leq \operatorname{weight} F \leq \mathfrak l\), which contradicts the fact that \(\operatorname{weight}\Phi \geq \mathfrak n\). The theorem is proved.

Proof of Theorem 2. Suppose the contrary. Put
\[ \mathfrak l=\max \left[\sup_\alpha \mathfrak m_\alpha,\ \sup_{x\in R}\chi(x,R)\right]. \]
If \(\mathfrak n\) is the least cardinal number greater than \(\mathfrak l\), then \(\operatorname{weight}R\geq \mathfrak n\). Let \(R=f(X)\) and let \(\Sigma_{\mathfrak l}^{p}\) be the \(\Sigma_{\mathfrak l}\)-product of the \(X_\alpha\) with base point \(p\). Consider \(Y=f(\Sigma_{\mathfrak l}^{p})\). By Theorem 1, \(Y\ne R\), for otherwise
\[ \operatorname{weight}R\leq \max\left(\mathfrak l,\sup_\alpha \mathfrak m_\alpha\right)\leq \mathfrak l, \]
which contradicts the supposition. Let \(x\in R\setminus Y\); then we shall show that \(\chi(x,R)\geq \mathfrak n>\mathfrak l\), and the theorem will be proved. Indeed, if \(\chi(x,R)\leq \mathfrak l\), then consider a fundamental system of neighborhoods \(\{O_\nu x\}\) at the point \(x\) of cardinality \(\mathfrak l\). Since \(Y\) is dense in \(R\), choose for each point \(z_\nu\in O_\nu x\cap Y\). Denote the set obtained by \(Z=\{z_\nu\}\). Note that \(\operatorname{card} Z\leq \mathfrak l\) and that \([Z]_R\ni x\). We shall show that this cannot be. For each point \(z_\nu\in Z\) choose a point \(t^\nu\in f^{-1}(z_\nu)\cap \Sigma_{\mathfrak l}^{p}\). Denote by \(T_\nu\) the set of all indices for which \(t_\alpha^\nu\ne p_\alpha\); since \(t^\nu\in \Sigma_{\mathfrak l}\), \(\operatorname{card}T_\nu\leq \mathfrak l\). Put
\[ T=\bigcup_\nu T_\nu. \]
Consider
\[ X_0=\prod_{\alpha\in T} X_\alpha. \]
This set may be regarded as a subspace of \(X\), containing all the points \(t^\nu\), by putting, for any \(x=(x_\alpha)\in X_0\), \(x_\alpha=p_\alpha\) if \(\alpha\in \Theta\setminus T\). At the same time, since \(\operatorname{card}T\leq \mathfrak l\cdot \mathfrak l=\mathfrak l\), we have \(X_0\subset \Sigma_{\mathfrak l}^{p}\). On the other hand, \(X_0\) is bicompact; therefore
\[ [\{t^\nu\}]_X\subset X_0\subset \Sigma_{\mathfrak l}^{p}. \]
Thus,
\[ f[\{t^\nu\}]_X=[Z]_R\subset f(\Sigma_{\mathfrak l}^{p})=Y, \]
which contradicts the fact that \([Z]_R\ni x\). The theorem is proved.

§ 2. Metrizability criteria for dyadic bicompacts.

Theorem 3. Let \(R\) be a dyadic bicompact. Then the following conditions are equivalent:

A. \(R\) is metrizable.
B. \(R\) is hereditarily normal.
C. The sequential closure of any subset in \(R\) coincides with the topological closure\(*\).

Let us note that from this theorem it follows that a dyadic bicompact satisfying the first axiom of countability is metrizable, since in such a bicompact the sequential closure of any subset coincides with the topological one. On the other hand, combining Theorem 4 from the author’s paper \((^8)\) and Theorem 3 from the paper \((^9)\), we obtain another criterion for metrizability of dyadic bicompacts.

Theorem 4. Let \(R\) be a dyadic bicompact. Then the following conditions are equivalent:

D. \(R\) is metrizable.
E. \(R\) is hereditarily dyadic with respect to closed sets.
F. In \(R\) there exists everywhere dense subset with the first axiom of countability.

We shall need the following lemma, which is also of independent interest.

Lemma 2**. Every nonisolated point of a dyadic bicompact is countably attainable***.

Proof. Let \(x\) be a nonisolated point of a dyadic bicompact \(R\). If \(\chi(x,R)=\aleph_0\), then \(x\) is countably attainable. If \(\chi(x,R)\geq \aleph_1\), then Theorem 4 \((^8)\) asserts that in \(R\) there lies \(Y=b_0E_{\aleph_1}\)—

\(*\) \(x\in[M]\) in the sequential topology \(R\), if there exists a countable sequence \([x_n]\in M\) converging to \(x\).

\(**\) As became known to me, an analogous result was obtained by M. Katetov.

\(***)\) A point \(x\) is countably attainable if in \(R\) there exists a countable sequence of pairwise distinct points converging to \(x\).

the minimal bicompact extension of a discrete space of cardinality $\aleph_1$—so that the only non-isolated point of the bicompactum $Y$ is the point $x$. However, it is easy to verify that the point $x$ is countably attainable for the bicompactum $Y$. Consequently, $x$ is countably attainable for $R$.

Proof of Theorem 3. Let us note that A follows from B and C; therefore, to prove the theorem it is sufficient to show that B implies A and C implies A.

1) B implies A. Let the weight $R \geqslant \aleph_1$; then $R=f(D^\tau)$, where $\tau \geqslant \aleph_1$. Let $\Sigma^p$ be the corresponding $\Sigma$-product of simple two-point spaces lying in $D^\tau$. Since $R$ is not metrizable, it follows that $Y=f\Sigma^p \ne R$. Consider some point $x \in R \setminus Y$. We shall prove that $R \setminus x$ is not normal. Note that $\Sigma^p$ is countably compact and, consequently, pseudocompact. It follows that $Y$ is pseudocompact; hence $R \setminus x$, containing $Y$, is also pseudocompact. If $R \setminus x$ were normal, then it would be countably compact. But this is not so, since the point $x$ is non-isolated and, consequently, countably attainable by Lemma 2.

2) C implies A. Suppose the contrary. Just as in the preceding proof, consider $Y=f(\Sigma^p)$, where $\Sigma^p \subset D^\tau$, $\tau \geqslant \aleph_1$. Since $R$ is not metrizable, $Y \ne R$. Note that $Y$ is dense in $R$ and, consequently, $[Y]_R=R$. We now show that the closure of $Y$ in the sequential topology coincides with $Y$. Indeed, by virtue of the countable compactness of $Y$, no countable sequence $\{y_n\}\in Y$ can converge to a point $x\in R\setminus Y$. The theorem is proved.

The author expresses sincere gratitude to A. Pelczynski, who carefully read this work and made a number of valuable comments.

Moscow State University
named after M. V. Lomonosov

Received
12 IV 1963

REFERENCES

1. H. H. Corson, Am. J. Math., 81, No. 3, 785 (1959).
2. Yu. M. Smirnov, Matem. sborn., 30, No. 1, p. 213 (1952).
3. M. H. Stone, Math. Mag., 21, 167 (1948).
4. P. S. Aleksandrov, UMN, 2, No. 1 (17), 5 (1947).
5. I. Glicksberg, Trans. Am. Math. Soc., 90, 369 (1959).
6. L. Gillman, M. Jerison, Rings of Continuous Functions, Princeton—N. Y., 1960.
7. E. Hewitt, Bull. Am. Math. Soc., 52, 641 (1946).
8. B. Efimov, DAN, 149, No. 5, 1011 (1963).
9. B. Efimov, DAN, 151, No. 5 (1963).