UDC 513.83+519.05

MATHEMATICS

B. EFIMOV

ON SUBSPACES OF DYADIC BICOMPACTS

(Presented by Academician P. S. Aleksandrov on 25 IV 1968)

The depth \(gX\) of a topological space \(X\) will mean the supremum of the cardinalities of closures of discrete subspaces \(T \subset X\). A. V. Arhangel’skii \((^1)\) posed the question: for what spaces is \(gX = |X|\)?* It is easy to construct examples of Hausdorff spaces \(X\) for which \(gX < |X|\). Here we shall give a scheme for proving that, under certain assumptions of set theory, the depth of every dyadic bicompact \(R\) is equal to its cardinality. Moreover, we shall show that \(gR\) is always attained, i.e. there exists a discrete \(T \subset R\) such that \(|\overline{T}| = |R|\). It is further shown that if \((wR)^{\aleph_0} = wR\), then \(R\) contains all extremally disconnected spaces of weight \(\leq wR\). These results follow from the following main theorem.

Main theorem. Let \(R\) be a dyadic bicompact of weight \(\tau\), and suppose that one of the following conditions is satisfied: \(1)\ \tau = \aleph_{\alpha+1}\) or \(2)\ \aleph_1 \leq cf(\tau) < \tau\). Then \(R\) contains a bicompact \(X \subset R\) which is continuously mapped onto \(D^\tau\).

Here \(cf(\tau)\) denotes the cofinal character of the number \(\tau\), the minimum of the cardinalities of sets \(A\) such that

\[ \tau = \sum_{\alpha \in A} \mathfrak n_\alpha \]

and \(\mathfrak n_\alpha < \tau\). The remaining terms and notation are taken from the author’s previous works \((^{2-4})\).

§ 1. A system \(\mathfrak B=\{U_\alpha\}\) consisting of open subsets of a space \(X\) will be called a local \(\pi\)-base of a set \(F \subset X\) if for every neighborhood \(OF\) there exists a \(U_\alpha \in \mathfrak B\) such that \(U_\alpha \subset OF\). The minimum of the cardinalities of local \(\pi\)-bases of \(F\) will be called the \(\pi\)-character of \(F\) and denoted by \(\pi\chi(F,X)\). If \(F\) is a point \(x \in X\), then \(\pi\chi(x,X)\) will be called the \(\pi\)-character of the point \(x \in X\). We note that the local characteristic of a space “\(\pi\)-character” is analogous to the integral weight characteristic “\(\pi\)-weight,” introduced by V. I. Ponomarev \((^5)\). Let \(f:Y\to X\) be a continuous mapping of the space \(Y\) onto \(X\), and let \(F \subset X\). Then by the \(\nu\)-character of the set \(F\) in \(X\) relative to \(Y\) and \(f\) (notation \(\nu(F,X,Y,f)\)) we shall mean the minimum of the neighborhood characters \(\chi(\Phi,Y)\) of sets \(\Phi \subset f^{-1}F\). In what follows we shall assume everywhere that \(X=R=f(D^\tau)\) is a dyadic bicompact. Therefore \(Y=D^\tau\), \(f\) is some mapping of \(D^\tau\) onto \(R\), and \(\nu(F,X,Y,f)=\nu(F,R,D^\tau,f)=\nu(F,R)\).

Lemma 1. Let \(f:D^\tau\to R\) be some mapping of \(D^\tau\) onto \(R\), and let \(F\) be a closed nowhere dense subset of \(R\), with \(\pi\chi(F,R)\leq \mathfrak n\) and \(wF\leq \mathfrak n\). Then there exists a point \(x\in F\) such that \(\nu(x,R)\leq \mathfrak n\).

Corollary 1. For any point of a dyadic bicompact \(R\) the inequality holds

\[ \nu(x,R) \leq \pi\chi(x,R) \leq \delta(x,R) \leq \chi(x,R) \leq wR, \tag{1} \]

where \(\delta(x,R)\) is the \(\delta\)-character of the point \(x\) in \(R\) (see \((^{3,4})\)).

Theorem 1. Let \(R\) be a dyadic bicompact and \(M\) an arbitrary subspace of \(R\). If \(\nu(x,R)\leq \mathfrak m \geq \aleph_0\) for all \(x\in M\), then \(w\overline{M}\leq \mathfrak m\).

The proof of this theorem is completely analogous to the proof of Theorem 14 from \((^2)\). We note that, by virtue of inequality (1), Theorem 1 remains

* \(|X|\) is the cardinality of the set \(X\); \(wX\) is the weight of the space \(X\).

valid if in its formulation the \(\nu\)-character is replaced by the \(\pi\chi\)-, \(\delta\)-, or \(\chi\)-character.

Corollary 2. Let
\[ E(\mathfrak n)=\{x\in R,\ \nu(x,R)\leq \mathfrak n\},\qquad F(\mathfrak n)=\overline{E(\mathfrak n)}. \]
If \(R\) is dyadic and \(\mathfrak n<wR\), then
\[ U=R\setminus F(\mathfrak n)\ne\varnothing . \]

§ 2. Preimages of \(D^\tau\) embedded in dyadic bicompacts

Theorem 2. Let \(R\) be a dyadic bicompactum and let \(\nu(x,R)\geq \mathfrak n\geq \aleph_0\) for all points \(x\in R\), and let there be a mapping \(f:D^\tau\to R\). Then there exists a closed subset \(F\subset R\) which is continuously mapped onto \(D^{\mathfrak n}\).

Proof. Denote by
\[ I^{\mathfrak n}=\prod_{\alpha\in B} I_\alpha \]
the Tikhonov product of the intervals \(I_\alpha=\{0\leq x\leq 1\}\). The generalized Cantor discontinuum
\[ D^{\mathfrak n}=\prod_{\alpha\in B} D_\alpha \]
may be regarded in a natural way as embedded in \(I^{\mathfrak n}\), if one assumes that \(D_\alpha=\{0,1\}\subset I_\alpha\). We shall construct a continuous mapping \(\varphi:R\to I^{\mathfrak n}\) such that \(\varphi(R)\supset D^{\mathfrak n}\). Then \(F=\varphi^{-1}(D^{\mathfrak n})\) will be the required set. We define the mapping \(\varphi\) by a system of mappings \(\{\varphi_\alpha\}\), with each \(\varphi_\alpha:R\to I_\alpha\) and \(\varphi_\alpha(R)\supset D_\alpha\). We shall construct this system by transfinite induction. Namely, we construct a system of pairs
\[ \omega_\alpha=\{A_\alpha^0,A_\alpha^1\},\qquad \alpha\in B, \]
of the bicompactum \(R\), consisting of disjoint closed sets of type \(G_\delta\) in \(R\) such that \(|B|=\mathfrak n\) and for any finite set of indices \(\alpha_1,\ldots,\alpha_s,\ \alpha_i\in B\), and the corresponding set of zeros and ones \(i_1,\ldots,i_s\), we have
\[ A_{\alpha_1}^{i_1}\cap\cdots\cap A_{\alpha_s}^{i_s}\ne\varnothing . \]
Then, for each pair \(\omega_\alpha\), define a continuous real-valued function \(\varphi_\alpha:R\to I_\alpha\) such that
\[ A_\alpha^0=\varphi^{-1}(0),\qquad A_\alpha^1=\varphi^{-1}(1). \]
The system of functions \(\{\varphi_\alpha\}\) determines in the natural way a mapping
\[ \varphi=\prod_{\alpha\in B}\varphi_\alpha \]
of the bicompactum \(R\) into \(I^{\mathfrak n}\) by the rule
\[ y=\varphi(x)=\{\varphi_\alpha(x)\}\in I^{\mathfrak n}. \]
Let
\[ x=\{i_\alpha\}\in D^{\mathfrak n},\qquad \alpha\in B; \]
then
\[ \varphi^{-1}(x)=\bigcap_{\alpha\in B} A_\alpha^{i_\alpha}. \]
But the latter intersection is nonempty in \(R\), since the family
\[ \{A_\alpha^{i_\alpha}\},\qquad \alpha\in B, \]
is centered (the \(\alpha\)’s are distinct!). Thus \(\varphi(R)\supset D^{\mathfrak n}\). Let us construct the required system of pairs \(\omega_\alpha=\{A_\alpha^0,A_\alpha^1\}\). For \(\omega_1=\{A_1^0,A_1^1\}\) take an arbitrary pair consisting of disjoint closed sets of type \(G_\delta\) in \(R\). Suppose that for all ordinal numbers \(\alpha\) less than some \(\beta<\Omega(\mathfrak n)\), pairs
\[ (A_1^0,A_1^1),\ldots,(A_\alpha^0,A_\alpha^1)\ldots,\qquad \alpha<\beta, \]
have been constructed such that: 1) \(A_\alpha^0\cap A_\alpha^1=\varnothing\); 2) \(A_\alpha^i\) are closed of type \(G_\delta\) in \(R\); 3) for any set \(\alpha_1,\ldots,\alpha_s,\ \alpha_i<\beta\), and any set \(i_1,\ldots,i_s\) of zeros and ones, we have
\[ A_{\alpha_1}^{i_1}\cap\cdots\cap A_{\alpha_s}^{i_s}\ne\varnothing . \]
Consider the mapping
\[ \psi=\prod_{\alpha<\beta}\varphi_\alpha,\qquad \psi:R\to I^{|\beta|}, \]
where \(|\beta|=|\{\alpha,\alpha<\beta\}|\). As we showed earlier, \(\psi(R)\supset D^{|\beta|}\). Moreover, since \(f_\alpha(x)=i\) (\(i=0,1\)) if and only if \(x\in A_\alpha^i\), we have
\[ \psi^{-1}(x)=\bigcap_{\alpha<\beta} A_\alpha^{i_\alpha},\qquad \text{if } x=\{i_\alpha\}\in D^{|\beta|}. \]
Put \(F_x=\psi^{-1}(x)\). Since
\[ wI^{|\beta|}=|\beta|, \]
there exists a bicompactum \(Y\subset R\) such that
\[ \psi(Y)=\psi(R) \]
and
\[ wY\leq |\beta| \]
(see \((6)\)). Thus
\[ Y\cap F_x\ne\varnothing \]
for all \(x\in D^{|\beta|}\). We shall show that there exists a neighborhood \(OY\) of the bicompactum \(Y\) in \(R\) such that
\[ (R\setminus OY)\cap F_x\ne\varnothing \]
for all \(x\in D^{|\beta|}\). Suppose the contrary. This means that for every neighborhood \(OY\) there exists \(x\in D^{|\beta|}\) such that \(F_x\subset OY\). By the bicompactness of \(R\) and the continuity of \(\psi\), there exists a basic open set \(U\subset\psi(R)\) such that
\[ x\in U \]
and
\[ \psi^{-1}U\subset OY. \]
Since
\[ w(\psi R)\leq wI^{|\beta|}=|\beta|, \]
it follows from this that
\[ \pi\chi(Y,R)\leq |\beta|. \]
On the other hand,
\[ wY\leq |\beta|, \]
and hence, by Lemma 1, we obtain that there exists a point \(y\in Y\) for which
\[ \nu(y,R)\leq |\beta|<\mathfrak n, \]
which contradicts the assumption. (We note that \(Y\) is nowhere

not dense, for otherwise for any point \(y\in \operatorname{int}Y\ne\varnothing\) we would have
\(\nu(x,R)\le uY\le |\beta|<\mathfrak n\). Thus there exists a neighborhood \(OY\) such that
\((R\setminus OY)\cap F_x\ne\varnothing\) for all \(x\in D^{|\beta|}\). Put \(Z=R\setminus OY\). Then \(Y\) and \(Z\) are disjoint closed sets, and, by normality of \(R\), there exists a real function \(\varphi_\beta\) such that
\(\varphi_\beta^{-1}(0)\supset Z\), \(\varphi_\beta^{-1}(1)\supset Y\), and
\(\varphi_\beta^{-1}(0)\cap\varphi_\beta^{-1}(1)=\varnothing\). Put
\(A_\beta^0=\varphi_\beta^{-1}(0)\), \(A_\beta^1=\varphi_\beta^{-1}(1)\). The pair
\(\omega_\beta=(A_\beta^0,A_\beta^1)\) is the desired one. Indeed, the centeredness of the system
\(\{A_\alpha^{i_\alpha}\}\), \(\alpha\le\beta\), for distinct \(\alpha\), follows from the fact that

\[ Y\cap\bigcap A_\alpha^{\alpha_i}\ne\varnothing \quad\text{and}\quad Z\cap\bigcap_{\alpha<\beta} A_\alpha^{i_\alpha}\ne\varnothing . \]

Theorem 2 is completely proved.

Corollary 3. If a dyadic bicompactum \(R\) contains a set \(F\) of type \(G_\delta\) (in particular, an open one), and moreover \(\nu(x,R)\ge \mathfrak n\ge \aleph_0\) for all \(x\in F\), then \(F\) (and, consequently, \(R\)) contains a closed \(X\) which is continuously mapped onto \(D^{\mathfrak n}\).

§ 3. Proof of the main theorem. 1) \(\tau=\aleph_{\alpha+1}\). Put \(\mathfrak n=\aleph_\alpha\); then, by Corollary 2, there exists a nonempty open set
\(U=R\setminus F(\mathfrak n)\subset R\) such that \(\nu(x,R)=\nu(x,U)\ge\tau\). Hence, by Corollary 3, there exists a closed \(X\subset R\) continuously mapped onto \(D^\tau\).

2) \(\aleph_1\le cf(\tau)<\tau\). In this case

\[ \tau=\sum_{\alpha\in A}\mathfrak n_\alpha,\quad \mathfrak n_\alpha<\tau \quad\text{and}\quad |A|=cf(\tau)=\mathfrak m<\tau . \]

Let \(F_\alpha=F(\mathfrak n_\alpha)\) be the sets defined in Corollary 2. Without loss of generality we shall assume that the sequence

\[ F_1\subset F_2\subset\cdots\subset F_\alpha\subset\cdots,\quad \alpha<\Omega(\mathfrak m), \]

is strictly increasing, well ordered by the uncountable regular ordinal \(\Omega(\mathfrak m)\). Then

\[ \Phi_1\subset \Phi_2\subset\cdots\subset\Phi_\alpha\subset\cdots,\quad \alpha<\Omega(\mathfrak m), \]

where \(\Phi_\alpha=f^{-1}F_\alpha\) and \(f:D^\tau\to R\) is some mapping of \(D^\tau\) onto \(R\), is an analogous sequence in \(D^\tau\). By a theorem of H. Shanin on calibers \((^7)\),

\[ L=\bigcap_{\alpha\in A}\left(D^\tau\setminus\Phi_\alpha\right)\ne\varnothing, \]

if \(A=\{\alpha,\alpha<\Omega(\mathfrak m)\}\).

Let \(x\in L\). For each \(\alpha\in A\) choose a basic neighborhood

\[ U_\alpha(x)=H_{\alpha_1\ldots \alpha_s}^{\,i_1\ldots i_s} \]

of the point \(x\) in \(D^\tau\) \((^2)\) such that
\(U_\alpha(x)\subset D^\tau\setminus\Phi_\alpha\). Then
\(\bigcap_{\alpha\in A}U_\alpha(x)\subset L\), but the intersection

\[ \bigcap_{\alpha\in A}U_\alpha(x)=H_w^{\,i(w)} \]

is a layer of \(D^\tau\) (see \((^2)\)), and moreover
\(|w|=|A|=\mathfrak m\); consequently, \(H_w^{\,i(w)}\) is homeomorphic to \(D^\tau\) \((^2)\), since \(\mathfrak m<\tau\). Thus,

\[ H_w^{\,i(w)}\subset L \quad\text{and}\quad f\!\left(H_w^{\,i(w)}\right)\subset \bigcap_{\alpha\in A}(R\setminus F_\alpha). \]

Note that

\[ f\!\left(H_w^{\,i(w)}\right)=X_0 \]

is a dyadic bicompactum, and for every point \(x\in X_0\) we have \(\nu(x,R)=\tau\). We shall prove that in fact \(\nu(x,X_0)=\tau\), if

\[ f:H_w^{\,i(w)}\to X_0 \]

is the restriction of the mapping \(f\) to \(H_w^{\,i(w)}\). Indeed, if there existed a point \(x_0\in X_0\) such that \(\nu(x_0,X_0)=\mathfrak n<\tau\), then there would exist a set \(Z\subset \tilde f^{-1}(x_0)\) and
\(\chi(Z,H_w^{\,i(w)})=\mathfrak n\). Then

\[ \chi(Z,D^\tau)\le \chi(Z,H_w^{\,i(w)})\cdot \chi(H_w^{\,i(w)},D^\tau) =\mathfrak n\cdot |w| =\mathfrak n\cdot \mathfrak m<\tau . \]

Thus \(\nu(x_0,R)=\mathfrak n\cdot\mathfrak m<\tau\), contrary to what was proved earlier. Hence there exists a dyadic bicompactum \(X_0\subset X\) such that
\(\nu(x,X_0)=\tau\) for all \(x\in X_0\). By Theorem 2, in \(X_0\) there lies a bicompactum \(X\) which is continuously mapped onto \(D^\tau\). The theorem is proved.

§ 4. Depth of images of the generalized Cantor discontinuum.

Lemma 2. If a bicompactum \(X\) is continuously mapped onto \(D^\tau\), then \(X\) topologically contains a discrete space \(T\) such that \(|\overline T|\ge 2^\tau\).

Proof. Let \(M\) be some dense subset of \(D^\tau\), \(|M|=\tau\), \(T\) a discrete space, \(|T|=\tau\), and \(\psi:M\to T\) an arbitrary one-to-one mapping of \(M\) onto \(T\). Let \(Y=D^\tau\oplus T\) be the disjoint union of \(D^\tau\) and \(T\). Introduce on \(Y\) the following topology. The points of the set \(T\) will be considered isolated. For any point \(x\in D^\tau\), declare a neighborhood \(U(x)\) to be the set

\[ U(x)=H(x)\cup \]

\(\bigcup \varphi(H(x)\cap M)\setminus \varphi(x)\), if \(H(x)\) is a basic neighborhood of \(x\) in \(D^\tau\). It can be shown that \(Y\) in this topology is a zero-dimensional compact extension of \(T\), moreover \(|Y|=2^\tau\) and \(wY=\tau\). Since, by N. B. Vedenisov’s theorem, \(D^\tau\) is universal for all zero-dimensional spaces, we have \(Y\subset_{\mathrm{top}}D^\tau\). Let \(f:X\to D^\tau\) be a continuous mapping of \(X\) onto \(D^\tau\). By Brouwer’s theorem \(({}^8), p. 178\), there exists an irreducible preimage \(Z\) of the space \(Y\) lying in \(X\). Since under an irreducible mapping the preimage of an isolated point is an isolated point, and the preimage of a dense set is dense, \(Z\) is a compact extension of \(T\). Since \(f(Z)=Y\), it follows that \(|Z|\ge 2^\tau\). The lemma is proved.

Theorem 3. Let \(R\) be a dyadic compactum of weight \(\tau\), with \(cf(\tau)=\aleph_0\). Then \(gR=|R|\). In particular, the depth of a metrizable compactum is equal to its cardinality and is attained.

Theorem 4. If the weight of a dyadic compactum \(R\) satisfies one of the conditions of the main theorem, then the depth of \(R\) is equal to its cardinality and is attained.

Theorem 5. The depth of any compactum without isolated points is not less than the continuum.

Theorem 6. Every dyadic compactum without isolated points contains a Cantor perfect set.

§ 5. Consider the following three conditions.

\((\zeta_1)\). For every ordinal number \(a\) we have \(2^{\aleph_a}=\aleph_{a+1}\).

\((\zeta_2)\). There are no weakly inaccessible cardinals.

\((\zeta_3)\). Every weakly inaccessible cardinal is also strongly inaccessible.

The definitions and properties of weakly and strongly inaccessible cardinal numbers we take from A. Tarski \(({}^9)\). The definitions and properties of absolutes of topological spaces we take from V. I. Ponomarev \(({}^5)\). Theorems proved using the conditions \((\zeta_1)\), \((\zeta_2)\), or \((\zeta_3)\) are marked respectively by the letters \(\zeta_1,\zeta_2,\zeta_3\).

Lemma 3 \((\zeta_1\) or \(\zeta_3)\). If the weight of a dyadic compactum \(R\) is a weakly inaccessible cardinal, then there exists in \(R\) a nonempty open set \(U\subset R\) such that \(v(x,R)=wR\) for all \(x\in U\) and for any mapping \(f:D^\tau\to R\).

The proof of this lemma is analogous to the proof of Theorem 16 in \(({}^2)\).

Theorem 7 \((\zeta_1\) or \(\zeta_2\) or \(\zeta_3)\). The depth of any dyadic compactum is equal to its cardinality and is attained.

The proof of this theorem follows from the main theorem, Theorems 4, 5, Lemmas 3 and 4, and Corollary 4.

Theorem 8 \((\zeta_1\) or \(\zeta_2\) or \(\zeta_3)\). Every dyadic compactum whose weight is an admissible cardinal, i.e. \((wR)^{\aleph_0}=wR\), topologically contains the absolute \(D^\tau\), and consequently \(({}^3)\), all extremally disconnected spaces of weight \(\le wR\).

Theorem 9 \((\zeta_1)\). The weight of a dyadic compactum \(R\) \((wR\ge \aleph_1)\) is equal to the least upper bound of the weights of Stone–Čech compactifications of discrete spaces lying in it. If \(wR=2^\tau,\ \tau\ge \aleph_0\), then the least upper bound is attained.

The author expresses sincere gratitude to P. S. Aleksandrov for his attention to this work.

Central Economic-Mathematical Institute
Academy of Sciences of the USSR

Received
19 IV 1968

REFERENCES

1. A. V. Arhangel’skii, DAN, 175, No. 4, 751 (1967).
2. B. Efimov, Trudy Mosk. matem. obshch., 14, 211 (1965).
3. B. Efimov, DAN, 172, No. 4, 771 (1967).
4. B. Efimov, DAN, 178, No. 3, 525 (1968).
5. V. I. Ponomarev, UMN, 21, No. 4, 101 (1966).
6. R. Engelking, A. Pełczyński, Colloq. Math., 11, No. 1, 55 (1963).
7. H. A. Shanin, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 24, 3 (1948).
8. F. Hausdorff, Set Theory, Moscow, 1937.
9. A. Tarski, H. Keisler, Fund. Math., 53, 225 (1964).