UDC 513.831 + 519.5

MATHEMATICS

B. EFIMOV

ON MAPPINGS OF ZERO-DIMENSIONAL BICOMPACTS

(Presented by Academician P. S. Aleksandrov on 30 III 1967)

In this note a sufficient condition is given under which a zero-dimensional bicompact \(R\) is mapped onto the generalized Cantor discontinuum \(D^\tau\).

A system \(\mathfrak B=\{\bar U_\alpha\}\), consisting of canonically closed sets, will be called a \(\delta\)-system of a closed set \(F\) of a topological space \(R\), if the following two conditions are satisfied: 1) for any finite collection \(\bar U_{\alpha_1},\ldots,\bar U_{\alpha_s}\) belonging to the system \(\mathfrak B\), we have
\[ \operatorname{int}\bar U_{\alpha_1}\cap\cdots\cap \operatorname{int}\bar U_{\alpha_s}\ne\varnothing; \]
2) \(F=\bigcap_\alpha \bar U_\alpha\) for all \(\bar U_\alpha\in\mathfrak B\).

We shall call a system \(\mathfrak B\) small if for any open \(V\supset F\) there exists \(\bar U_{\alpha_0}\in\mathfrak B\) such that \(F_0\subset \bar U_{\alpha_0}\subset V\). The minima of the cardinalities of (small) \(\delta\)-systems will be called the \(\delta\)-pseudocharacter (\(\delta\)-character) of the set \(F\) and will be denoted respectively by \(\delta_\psi(F,R)\) or \(\delta(F,R)\). We note that if \(R\) is bicompact, then \(\delta_\psi(F,R)=\delta(F,R)\) for any closed set \(F\subset R\). Further, if the character of a closed set \(F\subset R\) is denoted by \(\chi(F,R)\), then always \(\delta(F,R)\le \chi(F,R)\). However, since \(F\) need not belong to \(\operatorname{int}\bar U_\alpha\) for any \(\bar U_\alpha\in\mathfrak B\), even for bicompacts a strict inequality \(\delta(F,R)<\chi(F,R)\) is possible. Nevertheless the following proposition is valid:

If \(R=\prod X_\alpha\) is the topological product of compact metrizable spaces \(X_\alpha\), then for any closed \(F\subset R\) we have \(\delta(F,R)=\chi(F,R)\).

A limiting cardinal number \(\mathfrak m=\lim_\alpha \mathfrak n_\alpha\), \(\mathfrak n_\alpha<\mathfrak m\), will be called singular. If \(\mathfrak m=\aleph_\zeta\), then \(\zeta\) is a limiting ordinal number. The cardinal number \(\mathfrak m^+\), immediately following the singular number \(\mathfrak m\), will be called metasingular. Thus, \(\mathfrak m^+=\aleph_{\zeta+1}\). Every cardinal number \(\mathfrak n\) of the form \(\aleph_{\zeta+2}\) will be called simple. By \(\log \mathfrak m\) we shall denote the least cardinal number \(\mathfrak n\) such that \(2^{\mathfrak n}=\exp \mathfrak n\ge \mathfrak m\). The cardinality of a set \(A\) is denoted by \(|A|\). If \(\xi\) is an ordinal number, then \(|\xi|\) denotes the cardinality of the set of all ordinal numbers \(<\xi\). By \(wR\), \(sR\), and \(cR\) are denoted respectively the weight, the density, and the cellularity of \(R\). Theorems proved with the aid of the generalized continuum hypothesis are marked by the letters g.c.h.

Theorem 1 (g.c.h.). If for every point \(x\) of a zero-dimensional bicompact \(R\) the inequality \(\delta(x,R)\ge \tau\ge \aleph_0\) holds, then \(R\) is continuously mapped onto \(D^\tau\), if \(\tau\) is not a metasingular number, or onto \(D^{\log \tau}\), if \(\tau\) is a metasingular number.

The proof of this theorem is based on the following two lemmas.

Lemma 1. Let \(R\) be a bicompact and let for every point \(x\in R\) we have \(\delta(x,R)\ge \tau\ge \aleph_0\). Let \(F\) be a closed subset of \(R\) and \(\delta(F,R)\le \mathfrak m<\tau\). Then \(|F|\ge \exp\tau\).

Proof. We shall first show that for every point \(x\in F\) we have \(\chi(x,F)\ge \tau\). Indeed, if for some point \(x_0\in F\) we had \(\chi(x_0,F)\le \mathfrak n<\tau\), then
\[ x_0=\bigcap_{\alpha\in A} O_\alpha\cap F, \]
where \(O_\alpha\) are open sets in \(R\). For each \(\alpha\in A\) consider such a \(\bar U_\alpha\) that \(x_0\in \operatorname{int}\bar U_\alpha\subset\)

\(\subset O_\alpha\). In this case
\[ x_0=\bigcap_{\alpha\in A}\overline{U}_\alpha . \]
Since \(\delta(F,R)\leq \mathfrak m\), there exists a \(\delta\)-system \(\mathfrak B=\{\overline V_\alpha\}\) such that
\[ F=\bigcap_{\alpha\in B}\overline V_\alpha,\qquad |B|\leq \mathfrak m . \]
Thus we obtain
\[ x_0=\bigcap_{\alpha\in A}\overline U_\alpha\cap \bigcap_{\alpha\in B}\overline V_\alpha \]
and \(|A|+|B|=\mathfrak n+\mathfrak m<\tau\), while the family \(\{\overline U_\alpha\}\cup\{\overline V_\alpha\}\) is a \(\delta\)-system, since \(x_0\in \operatorname{int}\overline U_\alpha\) for all \(\alpha\in A\). Hence \(\delta(x_0,R)<\tau\), contrary to the assumption. Thus, for every point \(x\in F\) we have \(\chi(x,F)\geq \tau\). From the Čech–Pospíšil theorem \((^1)\) it follows that \(|F|\geq \exp\tau\). The lemma is proved.

Lemma 2. Let \(R\) be a zero-dimensional bicompactum, and suppose that for each point of it we have \(\delta(x,R)\geq \tau\geq \aleph_0\). Let \(\Phi\) be a closed subset of \(R\) and \(|\Phi|\leq \tau\). Further, let \(f:R\to D^{\mathfrak m}\) be a continuous mapping onto the generalized Cantor discontinuum of weight \(\mathfrak m\), with \(\mathfrak m^{\aleph_0}<\tau\). Then there exists a clopen neighborhood \(U'=U(\Phi)\) of the set \(\Phi\) such that \(U''=R-U'\) has the property
\[ U''\cap f^{-1}y\ne \varnothing \]
for all \(y\in D^{\mathfrak m}\).

Proof. Denote \(F_y=f^{-1}y\). Suppose the contrary. This means that for every clopen neighborhood \(U_\alpha(\Phi)=U_\alpha\) there is an \(F_y\subset U_\alpha\). Denote by \(V_\alpha\) the set of all \(F_y\) lying in \(U_\alpha\). Since the mapping \(f\) is continuous and closed, \(V_\alpha\) is open. Further, for every \(\alpha\) we have \(\overline V_\alpha\cap\Phi=\varnothing\), for otherwise we would find an \(F_y\subset U_\alpha\setminus\overline V_\alpha\), contradicting the definition of \(V_\alpha\). We now note that the number of sets of the form \(f^{-1}G_\alpha=\overline V_\alpha\) is equal to the number of all canonically closed sets \(\overline G_\alpha\) in \(D^{\mathfrak m}\), i.e. \(\mathfrak m^{\aleph_0}\) (see (2)). Consider the set of all such \(\overline V_\alpha\), and let \(\{\overline V_\alpha\}=\mathfrak B\). By what has just been said, \(|\mathfrak B|\leq \mathfrak m^{\aleph_0}<\tau\). We show that the system \(\mathfrak B\) is a \(\delta\)-system. Consider an arbitrary finite number of elements \(\overline V_{\alpha_1},\ldots,\overline V_{\alpha_s}\) belonging to \(\mathfrak B\); for each of them consider the corresponding \(U_{\alpha_1},\ldots,U_{\alpha_s}\). Let
\[ U_{\alpha_1\ldots\alpha_s}=U_{\alpha_1}\cap\cdots\cap U_{\alpha_s} \]
be a clopen set. Consequently, by the assumption, there exists \(F_y\subset U_{\alpha_1\ldots\alpha_s}\), and hence also an entire neighborhood of the form \(f^{-1}G\), where \(G\) is an open set in \(D^{\mathfrak m}\), but \(f^{-1}G\) belongs to \(V_{\alpha_1},\ldots,V_{\alpha_s}\), i.e.
\[ \operatorname{int}(\overline V_{\alpha_1}\cap\cdots\cap \overline V_{\alpha_s})\ne \varnothing . \]
Thus the system \(\mathfrak B\) is a \(\delta\)-system; consequently, for
\[ L=\bigcap_\alpha \overline V_\alpha \]
we have
\[ \delta_\Phi(L,R)=\delta(L,R)=|\mathfrak B|<\tau . \]
Applying Lemma 1, we obtain \(|L|\geq \exp\tau\). This contradicts the fact that \(L\subset \Phi\), and consequently \(|L|\leq |\Phi|\leq \tau\). The lemma is proved.

Proof of Theorem 1. We shall carry out the proof of this theorem by transfinite induction. Namely, we construct a system of binary coverings \(\omega_\alpha=\{A_\alpha^0,A_\alpha^1\}\), \(\alpha\in A\), of the bicompactum \(R\), such that \(|A|=\tau\) in the first case, or \(|A|=\log\tau\) in the second case, and: 1) \(A_\alpha^0\cup A_\alpha^1=R\); 2) \(A_\alpha^0\cap A_\alpha^1=\varnothing\); 3) for any set of indices \(\alpha_1,\ldots,\alpha_s\) and the corresponding set of zeros and ones \(i_1,\ldots,i_s\) we have
\[ A_{\alpha_1}^{i_1}\cap\cdots\cap A_{\alpha_s}^{i_s}\ne\varnothing . \]
In that case the system of characteristic functions \(\{\varphi_\alpha\}\): \(\varphi_\alpha(A_\alpha^0)=0\) and \(\varphi_\alpha(A_\alpha^1)=1\), defines a natural mapping \(f:R\to D^{|A|}\) by the rule
\[ f(x)=\{\varphi_\alpha(x)\}\in D^{|A|} \]
(the fact that the mapping is “onto” follows from the centeredness of \(\{A_\alpha^{i(\alpha)}\}\) for distinct \(\alpha\)). We shall construct such a system of coverings \(\{\omega_\alpha\}\) by transfinite induction. For \((A_1^0,A_1^1)\) we choose an arbitrary clopen disjoint pair covering \(R\). Suppose that the pairs
\[ (A_1^0,A_1^1),\ldots,(A_\alpha^0,A_\alpha^1),\ldots,\quad \alpha<\xi<\omega(\tau), \]
have been constructed, for all ordinal numbers \(\alpha\) less than some \(\xi<\omega(\tau)\), so that for all \(\alpha<\xi\) we have: 1) \(A_\alpha^0\cup A_\alpha^1=R\); 2) \(A_\alpha^0\cap A_\alpha^1=\varnothing\); 3) for all \(\alpha_1,\ldots,\alpha_s<\xi\) and any set \(i_1,\ldots,i_s\) of zeros or ones we have
\[ A_{\alpha_1}^{i_1}\cap\cdots\cap A_{\alpha_s}^{i_s}\ne\varnothing . \]
Using the distributivity of intersection with respect to union of sets, we obtain
\[ R=\bigcap_{\alpha<\xi}(A_\alpha^0\cup A_\alpha^1)=\bigcup_{i(\alpha)}\bigcap_{\alpha<\xi} A_\alpha^{i(\alpha)}=\bigcup_x F_x,\quad x\in D^{|\xi|}, \tag{1} \]

where \(F_x=\bigcap_{\alpha<\xi} A_\alpha^{\,i(\alpha)}\), and \(x\) is an arbitrary sequence of zeros and ones on the set of all ordinal numbers \(<\xi\). Moreover, the decomposition \(R\) by formula (1) is a continuous partition generated by the continuous mapping \(\varphi_\xi:R\to D^{|\xi|}\) according to the rule \(\varphi_\xi(F_x)=\{i_\alpha\}=x\in D^{|\xi|}\). Consequently, \(F_x=\varphi^{-1}(x)\). Consider the cardinal number \(|\xi|\). Since \(\xi<\omega(\tau)\), we have \(|\xi|<\tau\). We shall prove that, in the case where \(\tau\) is singular or regular, the induction step is possible for every \(\xi<\omega(\tau)\), while if \(\tau\) is metasingular, the induction step is possible only for \(\xi<\omega(\log\tau)\).

Case 1 (\(\tau\) singular). Consider in \(D^{|\xi|}\) a dense subset \(S\) of cardinality \(|S|=|\xi|=\aleph_\beta<\tau\). Choose, for each point \(x_s\in\varphi_\xi^{-1}(s)\), \(s\in S\), and put \(M=\bigcup_{s\in S} \overline{x_s}\). Then \(|M|\le \exp\exp\aleph_\beta=\aleph_{\beta+2}<\tau\) (\(\tau\) is singular!) and \(M\cap F_x\ne\varnothing\) for every \(x\in D^{|\xi|}\), which follows from the closedness of the mapping \(\varphi_\xi\) and the bicompactness of \(R\).

Case 2 (\(\tau=\aleph_{\beta+2}\) regular). Since \(|\xi|<\tau\), we have \(|\xi|\le\aleph_{\beta+1}\). Since \(\exp\aleph_\beta=\aleph_{\beta+1}\), the space \(D^{|\xi|}\) contains a dense subset \(S\) of cardinality \(\aleph_\beta\) (see (3)). Constructing, as in the first case, the set \(M\), we obtain that \(|M|\le \exp\exp\aleph_\beta=\aleph_{\beta+2}=\tau\).

Case 3 (\(\tau=\aleph_{\beta+1}\) and \(\beta=\lim\alpha\) is metasingular). In this case \(|\xi|<\log\tau=\aleph_\beta\), where \(\aleph_\beta\) is a singular cardinal. Thus, as in the first case, we choose the set \(S\) and construct the closed set \(M\subset R\).

Thus, in all three cases there exists in \(R\) a closed set \(M\) of cardinality \(\le\tau\) (or \(\log\tau\)), and moreover \(M\cap F_x\ne\varnothing\) for all \(x\in D^{|\xi|}\). We use Lemma 2. For this, note that \(|\xi|^{\aleph_0}<\tau\), since \(|\xi|<\tau\) (or \(\log\tau\)), and \(\tau\) is a singular or regular (or metasingular) cardinal. From Lemma 2 it follows that there exists a clopen neighborhood \(U^0(M)=U^0\) such that \(R\setminus U^0=U^1\) has the property that \(U^1\cap F_x\ne\varnothing\) for all \(x\in D^{|\xi|}\). At the same time note that \(F_x\cap U^0\ne\varnothing\) for all \(x\in D^{|\xi|}\), since \(U^0\supset M\). Next, it is obvious that \(U^0\cap U^1=\varnothing\) and \(U^0\cup U^1=R\). We take the pair \((U^0,U^1)\) as the required pair \((A_\xi^0,A_\xi^1)\). The induction step is completed.

We shall prove that property 3) is fulfilled. Let \(\alpha_1,\ldots,\alpha_s,\xi\) be an arbitrary set of indices \(\alpha_i\le\xi\), and \(i_1,\ldots,i_s,i_{s+1}\) an arbitrary set of zeros and ones. Note that
\[ A_{\alpha_1}^{i_1}\cap\cdots\cap A_{\alpha_s}^{i_s}\supset F_{x_0}, \]
where \(x_0\) is the point in \(D^{|\xi|}\) having, in the places \(\alpha_1,\ldots,\alpha_s\), the values \(i_1,\ldots,i_s\). But, by construction,
\[ F_{x_0}\cap(A_\xi^{i_{s+1}}=U^i)\ne\varnothing, \]
if \(i\) is equal to zero or one. Consequently,
\[ A_{\alpha_1}^{i_1}\cap\cdots\cap A_{\alpha_s}^{i_s}\cap A_\xi^{i_{s+1}}\ne\varnothing . \]
Property 3) is proved. Thus, using the principle of transfinite induction, for every \(\xi<\tau\) (or \(\log\tau\)) we construct the system of binary open coverings mentioned above. Hence, as was already noted, there follows the existence of a continuous mapping \(f:R\to D^\tau\) (or onto \(D^{\log\tau}\)). Theorem 1 is completely proved.

Theorem 2. If a bicompactum \(R\) is mapped continuously onto the generalized Cantor discontinuum \(D^\tau\), \(\tau\ge\aleph_0\), then \(R\) contains no fewer than \(\exp\tau\) points \(x\), at each of which \(\chi(x,R)\ge\tau\).

The proof of this theorem follows from the following Lemma 3, which is a special case of a more general theorem proved by I. Juhász \((^4)\).

Lemma 3. Let \(f:R\to D^\tau\) be a continuous mapping of a bicompactum \(R\) onto \(D^\tau\). Then for every \(x\in D^\tau\) there exists a point \(y\in f^{-1}x\) such that \(\chi(y,R)\ge\tau\).

I do not know whether in Theorem 2 one can replace \(\chi(x,R)\) by \(\delta(x,R)\). In any case, in Lemma 3 this cannot be done. Let us further note that in the formulation of Theorem 1 one cannot replace the inequality \(\delta(x,R)\ge\tau\) by the inequality \(\chi(x,R)\ge\tau\). This follows from the fact that for every \(\tau\ge\aleph_0\) there exist

there exists a zero-dimensional ordered bicompactum \(X\), and moreover \(\chi(x,X)\geq \tau\) for all \(x\in X\). If it were possible to map \(X\) onto \(D^\tau\), then, by the theorem of S. Mardešić and P. Papić \((^5)\), it would follow that \(D^\tau\) is metrizable, i.e. \(\tau\leq \aleph_0\).

Theorem 3 (GCH). Let \(R\) be a zero-dimensional bicompactum. If for every point \(x\in R\) the inequality \(\delta(x,R)\geq \tau\) holds, then \(R\) topologically contains all extremally disconnected spaces of weight \(\leq \tau\) (\(\tau\) nonmeasurable) or of weight \(\leq \log \tau\) (\(\tau\) measurable).

The proof of this theorem is based on Theorem 1 and on a theorem of V. I. Ponomarev \((^6)\) stating that every irreducible perfect preimage of an extremally disconnected space \(X\) is homeomorphic to \(X\).

Theorem 4 (GCH). Let \(R\) be an extremally disconnected bicompactum. Then:

1) If \(wR=\aleph_{\alpha+3}\), then \(R\) maps continuously onto \(D^{\aleph_\alpha}\).

2) If \(wR=\tau\) or \(\tau^+\) or \(\tau^{++}\), where \(\tau\) is a singular cardinal, then \(R\) maps continuously onto \(D^{\mathfrak m}\) for every \(\mathfrak m<\tau\).

3) If \(R\) is a bicompactum homogeneous with respect to character of weight \(\tau\), then \(R\) maps continuously onto \(D^{\log \tau}\) for every \(\tau\).

The proof of this theorem is based on Theorem 1 and on the following two lemmas.

Lemma 4 (A. Hajnal, I. Juhász \((^7)\)). Let \(R\) be a Hausdorff space and \(\mathfrak m\) an infinite cardinal number. If the set of those points \(x\in R\) for which \(\chi(x,R)\leq \mathfrak m\) has cardinality greater than \(\exp \mathfrak m\), then \(R\) contains more than \(\mathfrak m\) pairwise disjoint open subsets.

Lemma 5. Let \(R\) be an extremally disconnected bicompactum containing \(\mathfrak m\geq \aleph_0\) pairwise disjoint open sets. Then \(R\) maps continuously onto \(D^{\operatorname{exp}\aleph}\).

Corollary 1. Every extremally disconnected bicompactum homogeneous with respect to character, of weight \(\tau\), if \(\tau\) is singular, has cardinality \(\exp \tau\).

Corollary 2. Every complete Boolean algebra of large cardinality contains a free subalgebra of sufficiently large cardinality. The estimates of this cardinality are the same as in Theorem 4.

The author does not know whether the estimates of the weight of \(D^{\mathfrak n}\) in Theorem 4 can be improved.

Moscow Geological Prospecting Institute
named after S. Ordzhonikidze

Received
22 III 1967

REFERENCES

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