MATHEMATICS

B. EFIMOV

UNIMODULAR WEIGHT FUNCTIONS AND THE PROBLEM OF P. S. ALEXANDROV AND P. S. URYSOHN IN THE THEORY OF BICOMPACTS

(Presented by Academician P. S. Alexandrov, 9 V 1964)

§ 1. Weight characteristics and weight functions. As early as in the memoir (¹) P. S. Alexandrov and P. S. Urysohn showed that in the theory of topological spaces an important role is played by the so-called weight characteristics. Such are, for example, the weight \((wX)\) of a space \(X\) or its density \((sX)\)—the least cardinality of an everywhere dense subset of \(X\). In the present paper we study weight characteristics that assign to each topological space \(X\) a cardinal number \(\mathfrak n X\) satisfying the following five conditions:

\(Q_1.\) If \(X\) is homeomorphic to \(Y\), then \(\mathfrak n X = \mathfrak n Y\).

\(Q_2.\) If \(U\) is an open subset of \(X\), then \(\mathfrak n U \leqslant \mathfrak n X\).

\(Q_3.\) If \(X = U_1 \cup \cdots \cup U_k\) and each \(U_i\), \(i = 1,\ldots,k\), is open in \(X\), then
\[ \mathfrak n X \leqslant \mathfrak n U_1 + \cdots + \mathfrak n U_k . \]

\(Q_4.\) If
\[ X = \bigcup_{\alpha \in A} U_\alpha, \]
where each \(U_\alpha\) is open in \(X\) and all the \(U_\alpha\) are disjoint, then
\[ \mathfrak n X \geqslant \sum_{\alpha \in A} \mathfrak n U_\alpha . \]

\(Q_5.\) \(\mathfrak n X = 0\) if and only if \(X = \varnothing\).

Besides the aforementioned \(wX\) and \(sX\), these conditions are satisfied by such weight characteristics as the Kuratowski number \(cX\) (²), the proper weight and the calibers of N. A. Shanin (³, ⁴), the combinatorial weight of Yu. M. Smirnov (⁵), and many others. We note that among all weight characteristics defined on a topological space \(X\) there exists a least one. Such a characteristic is the Kuratowski number \(cX\). Indeed,
\[ cX = \sup_\alpha |\mathfrak A_\alpha|^*, \]
if \(\mathfrak A_\alpha\) is a certain system of disjoint nonempty open subsets of \(X\), the supremum being taken over all such systems. Put
\[ V_\alpha = \bigcup_{\beta \in B} U_\beta^\alpha, \]
if \(\mathfrak A_\alpha = \{U_\beta^\alpha\}\), \(\beta \in B\). Then, by conditions \(Q_4\) and \(Q_5\), we obtain
\[ \mathfrak n V_\alpha \geqslant \sum_{\beta \in B} \mathfrak n U_\beta^\alpha \geqslant |B| = |\mathfrak A_\alpha|. \]
Since \(V_\alpha \subset X\), by \(Q_2\) we have \(\mathfrak n X \geqslant \mathfrak n V_\alpha \geqslant |\mathfrak A_\alpha|\), and this inequality is valid for all \(\alpha\). Hence it follows that
\[ \mathfrak n X \geqslant \sup_\alpha |\mathfrak A_\alpha| = cX . \]

Let \(\mathfrak n X\) be some weight characteristic of the space \(X\) satisfying conditions \(Q_1\)–\(Q_5\). Let \(\mathfrak B = \{O_\alpha x\}\), \(\alpha \in A\), be some fundamental system of neighborhoods of a point \(x \in X\). The cardinal number
\[ \mathfrak n(x,X) = \inf_{\alpha \in A} \bigl(\mathfrak n O_\alpha x\bigr) \]
will be called the local characteristic of the point \(x\) in the space \(X\). For example, if \(\mathfrak n X = wX\), then \(\mathfrak n(x,X) = w(x,X)\) is called the local weight (⁶) of the point \(x\) in \(X\); if \(\mathfrak n X = sX\), then \(\mathfrak n(x,X) = s(x,X)\) is the local density, etc. It is easy to see that \(\mathfrak n(x,X)\) does not depend on the choice of the fundamental system of the point \(x\).

* The cardinality of a set \(X\) is denoted by \(|X|\).

By a weight function we shall mean, first, a function defined on the set of points of a topological space \(X\) and assigning to each point \(x\in X\) either the local characteristic \(\mathfrak n(x,X)\) or the neighborhood character \(\chi(x,X)\); second, a function defined on the set of closed subsets of the space \(X\) and assigning to each closed subset \(F\subset X\) either the weight characteristic \(\mathfrak n F\), or the neighborhood character \(\chi(F,X)\). From the whole class of weight functions defined on the set of points of a space \(X\), we single out the step weight functions. Namely, we shall call a function \(\mathfrak n(x)\), \(x\in X\), stepwise if in \(X\) there exists a disjoint system of open sets \(\{U_\alpha\}\), \(\alpha\in A\), such that \(\left[\bigcup_{\alpha\in A} U_\alpha\right]=X\) and on each \(U_\alpha\) the weight function is constant, i.e. \(\mathfrak n(x,X)=\mathfrak n_\alpha\) for all \(x\in U_\alpha\).

Theorem 1. The weight function \(\mathfrak n(x)\) of any local characteristic \(\mathfrak n(x,X)\) of a regular space \(X\) is stepwise.

Let us note that the weight function \(\chi(x)\) is not stepwise even for bicompacta. For example, the Stone–Čech compactification \(\beta Q\) of the rational points of the real line has everywhere dense sets \(M_1=Q\) and \(M_2=\beta Q\setminus Q\) such that \(\chi(x,\beta Q)\leq \aleph_0\) for \(x\in M_1\) and \(\chi(x,\beta Q)\geq \aleph_1\) for \(x\in M_2\).

A weight function \(\mathfrak n(x)\) defined on the set of points of a space \(X\), or on the set of its closed subsets, will be called modular if for every open \(U\subset X\) we have \(\mathfrak n[U]\leq \sup_x \mathfrak n(x,[U])\), \(x\in[U]\).

Theorem 2. The weight function \(\mathfrak n(x)\) of any local characteristic \(\mathfrak n(x,X)\), defined on a bicompactum \(X\), is modular.

A weight function \(\mathfrak n(x)\) defined on the set of points of a space \(X\) will be called unimodular if for every open \(U\subset X\) and every dense set \(M\subset U\) we have \(\mathfrak w[U]\leq \sup_{x\in M}\mathfrak n(x,[U])\).

Correspondingly, a weight function \(\mathfrak nF\), defined on the set of closed subsets of \(X\), will be called unimodular if for every open \(U\subset X\) we have \(\mathfrak w[U]\leq \sup_F \mathfrak nF\) for all \(F\subset[U]\).

Theorem 3. The weight function \(\chi(x)\), defined on a nondiscrete dyadic space, is unimodular.

Theorem 4. Every function of a weight characteristic, defined on the set of all closed subsets of a dyadic bicompactum \(R\), is unimodular.

Theorem 5. If the function \(\chi(x)\) on a regular topological space is unimodular and its range of values is discrete, then \(\chi(x)\) is a stepwise function.

The proof of Theorem 3 is easy to obtain by using two theorems of the author (Theorem 3 from \((^7)\) and Theorem 3 from \((^8)\)). The proof of Theorem 4 is not simple. First of all it is proved that if \(\mathfrak wR=\sup \chi(x,R)\), \(x\in R\), then for every infinite cardinal number \(\aleph_{p+1}\leq \chi(x_p,R)\) the space \(R\) topologically contains a discrete space \(T_p\) of cardinality \(\aleph_{p+1}\). Next, the unimodularity of the function \(c(F)\), \(F\subset R\), is established. Hence, by the minimality of the weight function \(c(F)\) among all weight functions \(\mathfrak n(F)\), the unimodularity of any weight function follows. In the proof of Theorem 5, Theorem 1 plays an essential role. Theorem 5 is a far-reaching generalization of Theorem 1 from \((^6)\).

§ 2. \(A\)-classes. In the memoir \((^1)\), P. S. Alexandroff and P. S. Urysohn posed a problem that has remained unsolved up to now: does there exist a bicompactum with the first axiom of countability and of cardinality greater than the continuum? We shall now construct two classes of bicompacta for which this problem has a positive solution. More precisely, we shall call a class of bicompacta \(\mathfrak B\) an \(A\)-class if, for every \(X\in\mathfrak B\), the assertion “\(X\) satisfies

“the first axiom of countability” entails that “the cardinality of \(X\) does not exceed the cardinality of the continuum \(c\).” We shall call an \(A\)-class complete if it contains all bicompacta of cardinality \(\leq c\).

First class. Let \(\mathfrak{S}=\{Y_\alpha\}\), \(\alpha\in B\), be the set of all bicompacta of weight \(\leq c\). Put

\[ Z=\prod_{\alpha\in B}Y_\alpha . \]

We shall say that a bicompactum \(R\) belongs to the first class if there exists a cardinal number \(\tau\) such that \(R\) is a continuous image of \(Z^\tau\).

We show that this class is an \(A\)-class. Let \(R=f(Z^\tau)\) and \(\sup \chi(x,R)\leq \aleph_0\), \(x\in R\). Since

\[ Z^\tau=\left(\prod_{\alpha\in B}Y_\alpha\right)^\tau =\prod_{\beta}\prod_{\alpha}Y_\alpha =\prod_{\alpha,\beta}Y_{\alpha,\beta}, \]

then, applying Theorem 2 from the author’s paper \({}^{(9)}\), we obtain

\[ wR\leq \max\left[\sup_{\alpha,\beta} wY_{\alpha,\beta},\ \sup_{x\in R}\chi(x,R)\right] =\max(c,\aleph_0)=c. \]

Let \(M\) be a dense subset of \(R\) and \(|M|\leq c\). Since every point \(x\in R\) is completely determined by a sequence \(\{x_n\}\subset M\) converging to it, we have \(|R|\leq |M|^{\aleph_0}\leq c^{\aleph_0}=c\), as required. The completeness of this class follows from its definition. It is just as easy to show that it is closed with respect to the direct product, taken in any number, and to continuous mappings.

Second class. Let

\[ I^\tau=\prod_{\alpha\in B} I_\alpha \]

be the product of the intervals \(I_\alpha=\mathfrak{E}\{0\leq x\leq 1\}\). Denote by \(\Sigma\) the subset of \(I^\tau\) consisting of those points \(x=(x_\alpha)\in I^\tau\) for which \(x_\alpha\ne0\) for at most a countable set of coordinates. We shall say that a bicompactum \(R\) belongs to the second class if there exist a cardinal number \(\tau\) and a closed set \(F\subset \Sigma\subset I^\tau\) such that \(R\) is a continuous image of \(F\).

We prove that this class is an \(A\)-class. Let \(R=f(F)\), \(F\subset \Sigma\subset I^\tau\), and \(\sup_{x\in R}\chi(x,R)\leq \aleph_0\). Suppose that \(|R|\geq \mathfrak{m}>c\), where \(\mathfrak{m}\) is the least cardinal number greater than \(c\). We shall then show that \(R\) contains \(b_0T\), the minimal compactification of a discrete space \(T\) of cardinality \(\mathfrak{m}\). This will mean that \(\chi(Z,R)\geq \mathfrak{m}\), if \(Z\) is the vertex of the bicompactum \(b_0T\). We proceed to the construction of \(b_0T\). Put \(F_x=f^{-1}x\). From each \(F_x\) choose a point \(q_x\). Denote the chosen set by \(M\). Since \(M\subset \Sigma\), for each point \(q_x\in M\) the set \(A_x\subset B\) of coordinates different from zero is at most countable. We shall call the set \(A_x\) the base of the point \(q_x\). Denote by \(S\) the system of all sets \(\{A_x\}\). Observe that there exists a quasi-disjoint subsystem\(^*\) \(\sigma\subset S\) of cardinality \(\geq \mathfrak{m}\). Indeed, otherwise the cardinality of every quasi-disjoint subsystem would be \(\leq c\); then, by Michael’s theorem \({}^{(10)}\), we would have \(|S|\leq c^{\aleph_0}=c\), since \(|A_x|\leq \aleph_0\). Further, for each \(A_x\) there exist only \(c^{\aleph_0}=c\) different points having \(A_x\) as their base. Hence finally \(|M|\leq |S|\cdot c=c\), contrary to the supposition. Let \(\sigma\) be a quasi-disjoint system of cardinality \(\geq \mathfrak{m}\). Two cases are possible: either \(\bigcap A_x=\mathfrak{E}\ne\varnothing\), or \(\bigcap A_x=\varnothing\). Since both are treated analogously, consider only the first case. Note that \(|\mathfrak{E}|\leq \aleph_0\), and therefore the number of all possible real-valued functions defined on \(\mathfrak{E}\) does not exceed \(c^{\aleph_0}=c\). Since \(|\sigma|\geq \mathfrak{m}\), there exists a subsystem \(\sigma_1\subset \sigma\) of cardinality \(\geq \mathfrak{m}\) such that, for all \(A_x\in\sigma_1\), the values of the points \(q_x\) on the set \(\mathfrak{E}\) coincide. Consider the set of points \(M'=\{q_x\}\subset M\), for which \(A_x\in\sigma_1\). Since any index \(\alpha\in B\) belongs either to one

\(^*\) A system \(S\) of sets is called quasi-disjoint if every point \(x\in\bigcap S\) belongs to at most one \(s\in S\) \({}^{(10)}\).

$A_x$, or all together, then the set $M'$ is discrete in the topology induced from $I^\tau$. On the other hand, the point $t=\{t_\alpha\}$, defined as follows: $t_\alpha=0$ if $\alpha\notin \mathscr E$, and $t_\alpha=q_{x\alpha}$ if $q_x$ is any point of $M'$, is the unique limit point of $M'$ in $I^\tau$. It is easy to see that $t\in\Sigma$; therefore, by the closedness of $F$, $t\in F$. Let $t\in F_x$. Remove from $M'$ the point $q_x$, if it is contained in $F_x\cap M'$. Thus a bicompactum $b_0T$, $|T|\geq m$, $b_0T\subset F$, has been constructed, which is mapped one-to-one and continuously onto $R$. Consequently, on $b_0T$ the mapping $f$ is a homeomorphism and $f(b_0T)$ is the desired bicompactum. This means, as was noted above, that $|R|\leq c$. Thus the assertion is completely proved.

Theorem 6. Every separable bicompactum (i.e. $sR\leq \aleph_0$) of the second class is metrizable.

A system of real-valued functions $L=\{f_\alpha\}$, defined on a completely regular space $X$, will be called pointwise countable if, for each point $x\in X$, the set of all functions $f_\alpha(x)\neq 0$ is at most countable.

Theorem 7. A bicompactum $R$ is topologically contained in $\Sigma$ if and only if there exists on $R$ a pointwise countable system of functions separating all points of $R$.

Consider the bicompacta $A_1$ and $A_2$ constructed by P. S. Aleksandrov and P. S. Uryson in (1). Since $sA_1=\aleph_0$, while $wA_1=c$, $A_1$ does not belong to the second class. On the other hand, the bicompactum $A_2$ belongs to this class, since there exists on it a pointwise countable system of functions separating all points. Hence it follows, first, that this class is not complete, and, second, that the first axiom of countability still does not imply metrizability for bicompacta of this class.

In conclusion, the author expresses gratitude to P. S. Aleksandrov for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
29 IV 1964

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